A
THEORY
OF
ORDINARY
P-ADIC
CURVES
A
Theory
of
Ordinary
p-adic
Curves
By
Shinichi
MOCHIZUKI*
Table
of
Contents
Table
of
Contents
Introduction
§0.
Statement
of
Main
Results
§1.
Review
of
the
Complex
Theory
Beltrami
Differentials
The
Beltrami
Equation
The
Series
Expansion
of
a
Quasiconformal
Function
Uniformization
of
Hyperbolic
Riemann
Surfaces
Uniformization
of
Moduli
Stacks
of
Hyperbolic
Riemann
Surfaces
Quasidisks
and
the
Bers
Embedding
The
Infinitesimal
Form
of
the
Modular
Uniformizations
Coordinates
of
Degeneration
The
Parabolic
Case
Real
Curves
§2.
Translation
into
the
p-adic
Case
Gunning’s
Theory
of
Indigenous
Bundles
The
Canonical
Coordinates
Associated
to
a
Kähler
Metric
The
Weil-Petersson
Metric
from
the
Point
of
View
of
Indigenous
Bundles
The
Philosophy
of
Kähler
Metrics
as
Frobenius
Liftings
The
Dictionary
Loose
Ends
Received
September
8,
1995.
Revised
October
2,
1996.
1991
Mathematics
Subject
Classification:
14H10,
14F30
*
Research
Institute
for
Mathematical
Sciences,
Kyoto
University,
Kyoto
606,
Japan
1
Chapter
I:
Crystalline
Projective
Structures
§0.
Introduction
§1.
Schwarz
Structures
Notation
and
Basic
Definitions
First
Properties
of
Schwarz
Structures
Crystalline
Schwarz
Structures
and
Monodromy
Correspondence
with
P
1
-bundles
Schwarz
Structures
and
Square
Differentials
Normalized
P
1
-bundles
with
Connection
The
Schwarzian
Derivative
§2.
Indigenous
Bundles
Basic
Definitions
and
Examples
First
Properties
Existence
and
de
Rham
Cohomology
Indigenous
Bundles
of
Restrictable
Type
§3.
The
Obstruction
to
Global
Intrinsicity
Introduction
of
Cohomology
Classes
Computation
of
the
Second
Chern
Class
The
Case
of
Dimension
One
Appendix:
Relation
to
the
Complex
Analytic
Case
Chapter
II:
Indigenous
Bundles
in
Characteristic
p
§0.
Introduction
§1.
FL-Bundles
Deformations
and
FL-Bundles
The
p-Curvature
of
an
FL-Bundle
§2.
The
Verschiebung
on
Indigenous
Bundles
The
Definition
of
The
Verschiebung
The
p-Curvature
of
an
Admissible
Indigenous
Bundle
The
Infinitesimal
Verschiebung
Differential
Criterion
for
Admissibility
2
§3.
Hyperbolically
Ordinary
Curves
Basic
Definitions
The
Totally
Degenerate
Case
The
Case
of
Elliptic
Curves:
The
Parabolic
Picture
The
Case
of
Elliptic
Curves:
The
Hyperbolic
Picture
The
Generic
Uniformization
Number
Chapter
III:
Canonical
Modular
Frobenius
Liftings
§0.
Introduction
§1.
Generalities
on
Ordinary
Frobenius
Liftings
Basic
Definitions
The
Uniformizing
Galois
Representation
The
Canonical
p-divisible
Group
Logarithms
of
Periods
Compatibility
of
Differentials
Canonical
Liftings
of
Points
in
Characteristic
p
Canonical
Multiplicative
Parameters
Canonical
Affine
Coordinates
The
Relationship
Between
Affine
and
Multiplicative
Parameters
§2.
Construction
of
the
Canonical
Frobenius
Lifting
Modular
Frobenius
Liftings
Indigenous
Sections
of
D
Frobenius
Invariant
Indigenous
Bundles
§3.
Applications
of
the
Canonical
Frobenius
Liftings
Canonical
Liftings
of
Curves
over
Witt
Vectors
Canonical
Affine
Coordinates
on
M
g,r
Topological
Markings
and
Uniformization
by
Quadratic
Differentials
Canonical
Multiplicative
Parameters
The
Case
of
Elliptic
Curves
Chapter
IV:
Canonical
Curves
§0.
Introduction
3
§1.
The
Canonical
Galois
Representation
Construction
and
Global
Properties
The
Horizontal
Section
over
the
Ordinary
Locus
The
Canonical
Frobenius
Lifting
over
the
Ordinary
Locus
§2.
The
Canonical
Log
p-divisible
Group
Log
p-divisible
Groups
at
Infinity
Construction
of
the
Canonical
Log
p-divisible
Group
Review
of
the
Theory
of
[Katz-Mazur]
§3.
The
Compactified
Canonical
Frobenius
Lifting
The
Canonical
Frobenius
Lifting
and
the
Canonical
Log
p-divisible
Group
Local
Analysis
at
Supersingular
Points
Global
Hecke
Correspondences
§4.
p-adic
Green’s
Functions
Compactified
Frobenius
Liftings
The
Height
of
a
Frobenius
Lifting
Admissible
Frobenius
Liftings
Geometric
Criterion
for
Canonicality
Chapter
V:
Uniformizations
of
Ordinary
Curves
§0.
Introduction
§1.
Crystalline
Induction
The
Crystalline-Induced
MF
∇
-object
The
Ring
of
Additive
Periods
The
Crystalline-Induced
Galois
Representation
Relation
to
the
Canonical
Affine
Coordinates
The
Parabolic
Case
§2.
Canonical
Objects
over
the
Stack
of
Multiplicative
Periods
The
Stack
of
Multiplicative
Periods
The
Canonical
Log
p-divisible
Group
The
Canonical
Frobenius
Lifting
Bibliography
Index
4
Introduction
§0.
Statement
of
Main
Results
The
goal
of
this
paper
is
to
present
a
theory
of
r-pointed
stable
curves
of
genus
g
over
p-adic
schemes
(for
p
odd),
which,
on
the
one
hand,
generalizes
the
Serre-Tate
theory
of
ordinary
elliptic
curves
to
the
hyperbolic
case
(i.e.,
2g
−
2
+
r
≥
1),
and,
on
the
other
hand,
generalizes
the
complex
uniformization
theory
of
hyperbolic
Riemann
surfaces
(reviewed
in
§1
of
this
introductory
Chapter)
due
to
Ahlfors,
Bers,
et
al.
to
the
p-adic
case.
We
begin
by
setting
up
the
necessary
algebraic
machinery:
that
is,
the
language
of
indigenous
bundles
(due
to
Gunning,
although
we
rephrase
Gunning’s
results
in
a
more
algebraic
form).
An
indigenous
bundle
is
a
P
1
-bundle
over
a
curve,
together
with
a
connection,
that
satisfy
certain
properties.
One
may
think
of
an
indigenous
bundle
as
an
algebraic
way
of
encoding
uniformization
data
for
a
curve.
We
then
study
the
p-curvature
of
indigenous
bundles
in
characteristic
p,
and
show
that
a
generic
r-pointed
stable
curve
of
genus
g
has
a
finite,
nonzero
number
of
distinguished
indigenous
bundles
(P,
∇
P
),
which
are
characterized
by
the
following
two
properties:
(1)
the
p-curvature
of
(P,
∇
P
)
is
nilpotent;
(2)
the
space
of
indigenous
bundles
with
nilpotent
p-curvature
is
étale
over
the
moduli
stack
of
curves
at
(P,
∇
P
).
We
call
such
(P,
∇
P
)
nilpotent
and
ordinary,
and
we
call
curves
ordinary
if
they
admit
at
least
one
such
nilpotent,
ordinary
indigenous
bundle.
If
a
curve
is
ordinary,
then
choos-
ing
any
one
of
the
finite
number
of
nilpotent,
ordinary
indigenous
bundles
on
the
curve
completely
determines
the
“uniformization
theory
of
the
curve”
–
to
be
described
in
the
following
paragraphs.
Because
of
this,
we
refer
to
this
choice
as
the
choice
of
a
p-adic
quasiconformal
equivalence
class
to
which
the
curve
belongs.
After
studying
various
basic
properties
of
ordinary
curves
and
ordinary
indigenous
bundles
in
characteristic
p,
we
then
consider
the
p-adic
theory.
Let
M
g,r
be
the
moduli
stack
of
r-pointed
stable
curves
of
genus
g
over
Z
p
.
Then
we
show
that
there
exists
a
ord
canonical
p-adic
(nonempty)
formal
stack
N
g,r
together
with
an
étale
morphism
ord
N
g,r
→
M
g,r
ord
such
that
modulo
p,
N
g,r
is
the
moduli
stack
of
ordinary
r-pointed
curves
of
genus
g,
together
with
a
choice
of
p-adic
quasiconformal
equivalence
class.
Moreover,
the
generic
ord
degree
of
N
g,r
over
M
g,r
is
>
1
(as
long
as
2g
−
2
+
r
≥
1
and
p
is
sufficiently
large).
It
is
ord
over
N
g,r
that
most
of
our
theory
will
take
place.
Our
first
main
result
is
the
following:
5
ord
Theorem
0.1.
Let
C
log
→
(N
g,r
)
log
(where
the
“log”
refers
to
canonical
log
stack
struc-
tures)
be
the
tautological
ordinary
r-pointed
stable
curve
of
genus
g.
Then
there
exists
a
ord
log
,
together
with
a
canonical
indigenous
bun-
canonical
Frobenius
lifting
Φ
log
N
on
(N
g,r
)
log
log
dle
(P,
∇
P
)
on
C
.
Moreover,
Φ
N
and
(P,
∇
P
)
are
uniquely
characterized
by
the
fact
(P,
∇
P
)
is
“Frobenius
invariant”
(in
some
suitable
sense)
with
respect
to
Φ
log
N
.
Moreover,
there
is
an
open
p-adic
formal
substack
C
ord
⊆
C
of
“ordinary
points”
of
ord
the
curve.
The
open
formal
substack
C
ord
⊆
C
is
dense
in
every
fiber
of
C
over
N
g,r
.
Also,
there
is
a
unique
canonical
Frobenius
lifting
log
ord
)
→
(C
log
)
ord
Φ
log
C
:
(C
which
is
Φ
log
N
-linear
and
compatible
with
the
Hodge
section
of
the
canonical
indigenous
log
bundle
(P,
∇
P
).
Finally,
Φ
log
C
and
Φ
N
have
various
functoriality
properties,
such
as
func-
toriality
with
respect
to
“log
admissible
coverings
of
C
log
”
and
with
respect
to
restriction
to
the
boundary
of
M
g,r
.
This
Theorem
is
an
amalgamation
of
Theorem
2.8
of
Chapter
III
and
Theorem
2.6
of
Chapter
V.
In
some
sense
all
other
results
in
this
paper
are
formal
consequences
of
the
above
Theorem.
For
instance,
Corollary
0.2.
The
Frobenius
lifting
Φ
log
N
allows
one
to
define
canonical
affine
local
coordinates
on
M
g,r
at
an
ordinary
point
α
valued
in
k,
a
perfect
field
of
characteristic
p.
These
coordinates
are
well-defined
as
soon
as
one
chooses
a
quasiconformal
equivalence
class
to
which
α
belongs.
Also,
at
a
point
α
∈
M
g,r
(k)
corresponding
to
a
totally
degenerate
curve,
Φ
log
N
defines
canonical
multiplicative
local
coordinates.
This
Corollary
follows
from
Chapter
III,
Theorem
3.8
and
Definition
3.13.
ord
Let
α
∈
N
g,r
(A),
where
A
=
W
(k),
the
ring
of
Witt
vectors
with
coefficients
in
a
ord
perfect
field
of
characteristic
p.
If
α
corresponds
to
a
morphism
Spec(A)
→
N
g,r
which
is
Frobenius
equivariant
(with
respect
to
the
natural
Frobenius
on
A
and
the
Frobenius
ord
lifting
Φ
log
N
on
N
g,r
),
then
we
call
the
curve
corresponding
to
α
canonical.
Let
K
be
the
quotient
field
of
A.
Let
GL
±
2
(−)
be
the
group
scheme
which
is
the
quotient
of
GL
2
(−)
by
{±1}.
ord
Theorem
0.3.
Once
one
fixes
a
k-valued
point
α
0
of
N
g,r
,
there
is
a
unique
canonical
ord
α
∈
N
g,r
(A)
that
lifts
α
0
.
Moreover,
if
a
curve
X
log
→
Spec(A)
is
canonical,
it
admits
(1)
A
canonical
dual
crystalline
(in
the
sense
of
[Falt],
§2)
Galois
repre-
sentation
ρ
:
π
1
(X
K
)
→
GL
±
2
(Z
p
)
(which
satisfies
certain
properties);
6
(2)
A
canonical
log
p-divisible
group
G
log
(up
to
{±1})
on
X
log
whose
Tate
module
defines
the
representation
ρ;
log
ord
(3)
A
canonical
Frobenius
lifting
Φ
log
)
→
(X
log
)
ord
over
the
X
:
(X
ordinary
locus
(which
satisfies
certain
properties).
Moreover,
if
a
lifting
X
log
→
Spec(A)
of
α
0
has
any
one
of
these
objects
(1)
through
(3)
(satisfying
various
properties),
then
it
is
canonical.
This
Theorem
results
from
Chapter
III,
Theorem
3.2,
Corollary
3.4;
Chapter
IV,
Theorem
1.1,
Theorem
1.6,
Definition
2.2,
Proposition
2.3,
Theorem
4.17.
The
case
of
curves
with
ordinary
reduction
modulo
p
which
are
not
canonical
is
more
ord
complicated.
Let
us
consider
the
universal
case.
Thus,
let
S
log
=
(N
g,r
)
log
;
let
f
log
:
X
log
→
S
log
be
the
universal
r-pointed
stable
curve
of
genus
g.
Let
T
log
→
S
log
be
the
finite
covering
(log
étale
in
characteristic
zero)
which
is
the
Frobenius
lifting
Φ
log
N
of
log
log
log
Theorem
0.1.
Let
P
→
S
be
the
inverse
limit
of
the
coverings
of
S
which
are
log
log
log
log
log
log
iterates
of
the
Frobenius
lifting
Φ
log
.
Let
X
=
X
×
T
;
X
=
X
×
.
log
S
S
log
P
N
T
P
We
would
like
to
consider
the
arithmetic
fundamental
groups
def
def
Π
1
=
π
1
((X
T
log
)
Q
p
);
Π
∞
=
π
1
((X
P
log
)
Q
p
)
Unlike
the
case
of
canonical
curves,
we
do
not
get
a
canonical
Galois
representation
of
Π
1
into
GL
±
2
(Z
p
).
Instead,
we
have
the
following
Theorem
0.4.
There
is
a
canonical
Galois
representation
ρ
∞
:
Π
∞
→
GL
±
2
(Z
p
)
Moreover,
the
obstruction
to
extending
ρ
∞
to
Π
1
is
nontrivial
and
is
measured
precisely
by
the
extent
to
which
the
canonical
affine
coordinates
(of
Corollary
0.2)
are
nonzero.
Also,
log
there
is
a
ring
D
S
Gal
with
a
continuous
action
of
π
1
(T
Q
)
such
that
we
have
a
canonical
p
dual
crystalline
representation
Gal
ρ
1
:
Π
1
→
GL
±
2
(D
S
)
(i.e.,
this
is
a
twisted
homomorphism,
with
respect
to
the
action
of
Π
1
(acting
through
log
))
on
D
S
Gal
).
Finally,
the
ring
D
S
Gal
has
an
augmentation
D
S
Gal
→
Z
p
which
is
π
1
(T
Q
p
Π
∞
-equivariant
(for
the
trivial
action
on
Z
p
)
and
which
is
such
that
after
restricting
to
Π
∞
,
and
base
changing
by
means
of
this
augmentation,
ρ
1
reduces
to
ρ
∞
.
7
This
follows
from
Chapter
V,
Theorems
1.4
and
1.7.
All
along,
we
note
that
when
one
specializes
the
theory
to
the
case
of
elliptic
curves,
one
recovers
the
familiar
classical
theory
of
Serre-Tate.
For
instance,
the
definitions
of
“ordinary
curves”
and
“canonical
liftings”
specialize
to
the
objects
with
the
same
names
in
Serre-Tate
theory.
The
p-adic
canonical
coordinates
on
the
moduli
stack
M
g,r
(Corollary
0.2)
specialize
to
the
Serre-Tate
parameter.
The
Galois
obstruction
to
extending
ρ
∞
to
a
representation
of
Π
1
specializes
to
the
obstruction
to
splitting
the
well-known
exact
sequence
of
Galois
modules
that
the
p-adic
Tate
module
of
an
ordinary
elliptic
curve
fits
into.
For
more
detailed
accounts
of
the
results
in
each
Chapter,
we
refer
to
the
introductory
sections
at
the
beginnings
of
each
of
the
Chapters.
In
the
rest
of
this
introductory
Chapter,
we
explain
the
relationship
between
the
p-adic
case
and
the
classically
known
complex
case.
Acknowledgements:
I
would
like
to
thank
Prof.
Barry
Mazur
of
Harvard
University
for
providing
the
stimulating
environment
(during
the
Spring
of
1994)
in
which
this
paper
was
written.
Also,
I
would
like
to
thank
both
Prof.
Mazur
and
Prof.
Yasutaka
Ihara
(of
RIMS,
Kyoto
University)
for
their
efforts
in
assisting
me
to
publish
this
paper,
and
for
permitting
me
to
hold
lecture
series
at
Harvard
(Spring
of
1994)
and
RIMS
(Fall
of
1994),
respectively,
during
which
I
discussed
the
contents
of
this
paper.
Finally,
I
would
like
to
thank
Prof.
Ihara
for
informing
me
of
the
theory
of
[Ih],
[Ih2],
[Ih3],
and
[Ih4].
This
theory
anticipates
many
aspects
of
the
theory
of
the
present
paper
(especially,
the
discussion
of
Frobenius
liftings
and
pseudo-correspondences
in
Chapters
III
and
IV).
On
the
other
hand,
the
techniques
and
point
of
view
of
Prof.
Ihara’s
theory
differ
substantially
from
those
of
the
present
paper.
Moreover,
from
a
rigorous,
mathematical
point
of
view,
the
main
results
of
Prof.
Ihara’s
theory
neither
imply
nor
are
implied
by
the
main
results
of
the
present
paper.
However,
it
is
the
author’s
subjective
opinion
that
philosophically,
the
motivation
behind
Prof.
Ihara’s
theory
was
much
the
same
as
that
of
the
author’s.
§1.
Review
of
the
Complex
Theory
In
order
to
explain
the
meaning
of
the
main
results
of
this
paper,
it
is
first
necessary
to
review
the
complex
theory
of
uniformization
in
a
fashion
that
makes
the
generalization
to
finite
primes
more
natural.
This
is
the
goal
of
the
present
Section.
Since
all
of
the
material
is
“standard”
and
“well-known,”
we
shall,
of
course,
omit
proofs,
instead
citing
references
for
major
results.
We
shall
say
that
a
Riemann
surface
X
is
of
finite
type
if
it
can
be
obtained
by
removing
a
finite
number
of
points
p
1
,
.
.
.
,
p
r
from
a
compact
Riemann
surface
Y
of
genus
g.
Note
that
in
this
case,
Y
and
{p
1
,
.
.
.
,
p
r
}
are
uniquely
determined
up
to
isomorphism.
We
shall
say
that
the
Riemann
surface
of
finite
type
X
is
hyperbolic
(respectively,
parabolic;
elliptic)
if
2g
−
2
+
r
≥
1
(respectively,
2g
−
2
+
r
=
0;
2g
−
2
+
r
<
0).
In
this
paper,
we
shall
be
concerned
exclusively
with
Riemann
surfaces
of
finite
type
(and
their
uniformizations).
This
is
because
it
is
precisely
these
Riemann
surfaces
which
correspond
to
algebraic
objects.
Also,
we
shall
mainly
be
concerned
with
8
the
hyperbolic
case,
since
this
is
the
most
difficult.
Indeed,
from
the
point
of
view
of
the
theory
of
uniformization
and
moduli,
the
elliptic
case
is
completely
trivial,
and
the
parabolic
case
(although
nontrivial)
is
relatively
easy
and
explicit.
In
some
sense,
the
theme
of
our
review
of
the
classical
complex
theory
is
that
in
most
cases,
there
are
two
ways
to
approach
results:
the
“classical”
and
the
“quasiconformal.”
Typically,
the
classical
approach
was
known
earlier,
and
is
more
geometric
and
intuitive.
On
the
other
hand,
the
classical
approach
has
the
drawback
of
producing
theories
and
results
that
are
only
real
analytic,
rather
than
holomorphic
in
nature.
By
contrast
the
quasiconformal
approach,
which
was
pioneered
by
Ahlfors
and
Bers,
tends
to
give
rise
to
holomorphic
structures
and
results
naturally.
It
is
thus
natural
that
the
connection
between
the
“quasiconformal
approach”
and
the
p-adic
theory
should
be
much
more
natural
and
transparent.
Beltrami
Differentials
Let
X
be
a
Riemann
surface
(not
necessarily
of
finite
type).
Let
us
consider
the
complex
line
bundle
τ
X
⊗
ω
X
on
X,
where
ω
X
is
the
complex
conjugate
bundle
to
the
canonical
bundle
ω
X
,
and
τ
X
is
the
tangent
bundle.
Note
that
if
s
is
a
section
of
τ
X
⊗
ω
X
over
X,
then
we
can
consider
its
L
∞
-norm
s
∞
,
since
the
transition
functions
of
τ
X
⊗
ω
X
have
complex
absolute
value
1.
A
Beltrami
differential
μ
on
X
is
a
measurable
section
of
the
line
bundle
τ
X
⊗
ω
X
such
that
μ
∞
<
1.
Why
the
bundle
τ
X
⊗
ω
X
?
The
reason
is
that
this
bundle
is
closely
connected
with
the
moduli
of
the
Riemann
surface
X.
Indeed,
Let
us
consider
an
arbitrary
C
∞
section
μ
of
τ
X
⊗
ω
X
.
Now
since
τ
X
has
the
structure
of
a
holomorphic
line
bundle,
we
have
a
∂
operator
on
τ
X
.
If
we
look
at
global
C
∞
sections,
this
gives
us
a
complex
∂
C
∞
(X,
τ
X
)
−→
C
∞
(X,
τ
X
⊗
ω
X
)
which
computes
the
analytic
cohomology
of
τ
X
.
If
X
is,
for
instance,
compact,
then
this
analytic
cohomology
coincides
with
the
cohomology
in
the
Zariski
topology
of
the
algebraic
tangent
bundle.
Thus,
for
X
compact
and
hyperbolic,
the
above
complex
has
cohomology
groups
H
0
=
0,
and
H
1
=
H
1
(X,
τ
X
),
which
is
well-known
to
be
the
space
of
infinitesimal
deformations
of
X.
Moreover,
if
X
is
compact
of
genus
g
≥
2,
and
M
g
is
the
moduli
stack
of
curves
of
genus
g,
then
H
1
(X,
τ
X
)
is
precisely
the
tangent
space
to
M
g
at
the
point
defined
by
X.
At
any
rate,
(for
X
arbitrary)
we
have
a
natural
surjection
9
C
∞
(X,
τ
X
⊗
ω
X
)
→
H
1
(X,
τ
X
)
Thus,
the
image
of
μ
under
this
surjection
defines
an
infinitesimal
deformation
of
the
complex
structure
of
X.
This
establishes
the
relationship
between
sections
of
τ
X
⊗
ω
X
and
the
moduli
of
X.
The
reason
for
considering
measurable,
rather
than
just
C
∞
,
sec-
tions
is
that
it
is
easier
to
obtain
solutions
to
a
certain
differential
equation,
the
Beltrami
equation,
when
one
works
in
this
greater
generality.
The
Beltrami
Equation
Having
established
the
relationship
between
sections
of
τ
X
⊗
ω
X
and
infinitesimal
deformations,
we
now
would
like
to
integrate
–
i.e.,
to
“give
a
reciprocity
law”
–
that
assigns
to
a
section
μ
of
τ
X
⊗
ω
X
not
just
an
infinitesimal
deformation
of
X,
but
an
actual
new
Riemann
surface,
i.e.,
a
new
complex
structure
on
the
topological
manifold
underlying
X.
To
do
this,
we
consider
the
Beltrami
equation
∂f
=
μ
·
∂f
which
we
regard
as
a
differential
equation
in
the
unknown
function
f
.
It
is
a
nontrivial
result
(proven,
for
instance,
in
[Lehto2])
that
when
μ
is
a
Beltrami
differential,
there
exist
local
L
2
solutions
f
to
the
Beltrami
equation
that
are
homeomorphisms
(where
they
are
defined).
Such
functions
f
are
called
quasiconformal
(with
dilatation
μ).
If
f
and
g
(defined
on
some
open
set
U
⊆
X)
are
both
quasiconformal
with
the
same
dilatation
μ,
then
it
is
easy
to
see
that
∂
applied
to
f
◦
g
−1
(in
the
distributional
sense)
is
zero.
That
is,
f
=
h
◦
g
for
some
biholomorphic
function
h.
Thus,
up
to
composition
with
a
biholomorphic
function,
quasiconformal
solutions
to
the
Beltrami
equation
are
unique.
With
these
observations,
we
can
define
a
new
complex
structure
on
X
associated
to
a
Beltrami
differential
μ
as
follows.
Let
us
call
the
resulting
Riemann
surface
X
μ
.
Thus,
the
underlying
topological
manifold
of
X
μ
is
the
same
as
that
of
X.
On
an
open
set
U
⊆
X,
we
take
a
local
quasiconformal
function
f
of
dilatation
μ,
and
define
it
to
be
a
holomorphic
function
on
X
μ
.
By
the
essential
uniqueness
of
solutions
to
the
Beltrami
equation,
everything
is
well-defined,
and
so
we
obtain
a
new
global
Riemann
surface
X
μ
.
Thus,
the
assignment
μ
→
X
μ
is
the
fundamental
“reciprocity
law”
that
we
are
looking
for.
10
The
Series
Expansion
of
a
Quasiconformal
Function
In
order
to
really
understand
the
Beltrami
equation,
it
is
useful
to
look
at
the
explicit
representation
of
its
solutions
as
series
“in
μ”
(as
in
[Lehto],
pp.
25-27).
We
begin
by
considering
Cauchy’s
integral
formula:
1
f
(z)
=
2πi
1
f
(ζ)
dζ
−
ζ
−
z
π
∂D
∂f
(ζ)
dξdη
ζ
−
z
D
for
a
function
f
with
L
1
derivatives
on
an
open
disk
D
in
the
complex
plane.
Thus,
if
f
(and
its
L
1
derivatives)
are
defined
on
all
of
C,
and
f
(z)
→
0
as
z
→
∞,
then
we
obtain
f
(z)
=
T
∂f
where
T
is
the
operator
on
C
∞
functions
ω
with
compact
support
given
by
1
(T
ω)(z)
=
−
π
ω(ζ)
dξdη
ζ
−
z
C
Put
another
way,
(from
the
point
of
view
of
the
theory
of
pseudodifferential
operators)
T
is
the
parametrix
for
the
elliptic
differential
operator
∂.
If
we
define
the
Hilbert
transfor-
mation
H
by
ω(ζ)
dξdη
(ζ
−
z)
2
C
1
(H
ω)(z)
=
−
π
then
we
obtain
that
∂T
=
H.
Also,
it
can
be
shown
that
∂
and
∂
commute
with
both
T
and
H.
Now
let
us
suppose
that
μ
is
a
Beltrami
differential
on
C
(say,
with
compact
support),
and
that
f
is
quasiconformal
on
C
with
dilatation
μ.
Then
f
is
holomorphic
at
infinity,
and
so,
after
normalization,
in
a
neighborhood
of
infinity,
it
looks
like
f
(z)
=
z
+
b
n
z
−n
n≥1
for
some
b
n
∈
C.
Thus,
f
(z)
−
z
goes
to
0
as
z
→
∞,
so
we
obtain
that
∂f
(z)
=
1
+
∂{f
(z)
−
z}
=
1
+
∂T
∂{f
(z)
−
z}
=
1
+
H∂f
(z)
11
Thus,
since
∂f
=
μ
·
∂f
,
it
follows
that
∂f
=
μ
+
μ
·
H∂f
This
integral
equation
has
the
formal
solution
∂f
=
(μ
·
H)
i
μ
i≥0
which
converges
in
L
2
because
(1)
it
can
be
shown
that
H
extends
to
an
isometry
L
2
→
L
2
;
(2)
since
μ
is
a
Beltrami
differential,
μ
∞
<
1
(which
thus
explains
this
part
of
the
definition
of
a
Beltrami
differential).
Thus,
applying
the
operator
T
,
we
get
the
series
solution
f
(z)
=
z
+
T
{
(μ
·
H)
i
μ}
i≥0
to
the
Beltrami
equation.
From
our
point
of
view,
this
series
solution
has
two
important
consequences.
First
of
all,
the
set
of
all
possible
μ
clearly
form
an
open
subset
of
a
(rather
large)
complex
vector
space
(i.e.,
the
space
of
measurable
sections
of
τ
X
⊗
ω
X
).
Thus,
relative
to
the
complex
structure
of
this
complex
vector
space,
the
series
solution
makes
it
clear
that
f
depends
holomorphically
on
μ.
Second,
it
computes
the
infinitesimal
change
in
f
as
μ
varies
to
first
def
order.
Namely,
this
term
is
given
by
φ
=
T
(μ).
Note
that
∂φ
=
μ
It
turns
out
that
this
result
–
that
∂
applied
to
the
infinitesimal
change
φ
in
the
solution
to
the
Beltrami
equation
gives
us
back
μ
–
holds
for
arbitrary
Beltrami
differentials
μ.
(See,
e.g.,
[Gard],
p.
72).
The
reason
why
this
observation
is
interesting
is
as
follows.
Suppose,
for
simplicity,
that
μ
is
C
∞
.
Let
U
be
an
open
covering
of
X
such
that
the
intersection
of
any
finite
collection
of
open
sets
in
U
is
Stein.
Then
by
considering
the
standard
isomorphism
between
the
Čech
cohomology
(with
respect
to
U
)
and
the
∂-cohomology
of
τ
X
,
it
thus
follows
that
the
infinitesimal
deformation
X
·μ
(where
is
“small”)
in
the
complex
structure
of
X
given
by
solving
the
Beltrami
equation
is
precisely
the
same
as
the
infinitesimal
deformation
given
by
mapping
μ
to
H
1
(X,
τ
X
)
via
the
surjection
12
C
∞
(X,
τ
X
⊗
ω
X
)
→
H
1
(X,
τ
X
)
considered
previously.
This
completes
the
justification
of
the
claim
that
the
assignment
μ
→
X
μ
is
an
“integrated
version”
of
the
“infinitesimal
reciprocity
law”
C
∞
(X,
τ
X
⊗
ω
X
)
→
H
1
(X,
τ
X
)
that
follows
just
from
the
definition
of
the
∂-cohomology
of
τ
X
.
Uniformization
of
Hyperbolic
Riemann
Surfaces
be
its
universal
covering
space.
Thus,
Let
X
be
a
hyperbolic
Riemann
surface.
Let
X
X
inherits
a
natural
complex
structure
from
X.
Then
one
of
the
most
basic
results
in
the
field
is
that
we
have
an
isomorphism
of
Riemann
surfaces
∼
X
=
H
where
H
is
the
upper
half
plane.
By
considering
the
covering
transformations
of
H
→
X,
we
get
a
homomorphism
(well-defined
up
to
conjugation)
ρ
:
π
1
(X)
→
Aut(H)
⊆
PSL
2
(R)
which
we
call
the
canonical
representation
of
X.
There
are
(at
least)
two
ways
to
prove
this
result.
The
first
approach
is
the
classical
approach,
and
goes
back
to
Koebe’s
work
in
the
early
twentieth
century.
It
involves
There
is
an
intrinsic,
a
priori
definition
of
considering
Green’s
functions
G(−,
−)
on
X.
Green’s
functions,
which
is
not
important
for
us
here.
A
posteriori,
that
is,
once
one
knows
∼
that
X
=
H,
we
can
pull-back
the
hyperbolic
metric
dx
2
+
dy
2
y
2
so
that
we
obtain
a
hyperbolic
distance
function
on
X.
Then
G(x,
y)
(for
on
H
to
X,
is
given
by
the
logarithm
of
the
hyperbolic
distance
between
x
and
y.
One
can
x,
y
∈
X)
find
a
detailed
exposition
of
this
approach
in
[FK].
The
second
approach
(which
is
more
relevant
to
the
p-adic
case)
is
the
approach
of
Bers
([Bers]).
Suppose
that
X
is
obtained
by
removing
r
points
from
a
compact
Riemann
surface
Y
of
genus
g.
Then
one
first
observes
that
there
exists
a
Riemann
surface
X
which
is
obtained
by
removing
r
points
from
a
compact
Riemann
surface
of
genus
g
and
is
isomorphic
to
H.
Then
one
constructs
(from
purely
whose
universal
covering
space
X
13
elementary
considerations)
a
quasiconformal
homeomorphism
X
∼
=
X.
This
quasiconfor-
mal
homeomorphism
defines
a
Beltrami
differential
μ
on
X
,
which
we
can
pull
back
to
∼
X
=
H
to
obtain
a
Beltrami
differential
μ
H
on
H.
By
reflection,
one
extends
μ
H
to
a
Beltrami
differential
μ
on
C.
Then
we
solve
the
Beltrami
equation
for
μ
on
C
so
that
we
obtain
a
quasiconformal
homeomorphism
f
:
C
→
C
μ
which
goes
to
infinity
at
infinity.
Let
Γ
be
the
group
of
Möbius
transformation
of
H
over
X
.
Thus,
H/Γ
∼
defined
by
the
covering
transformations
of
X
=
X
.
Then
it
follows
from
the
uniqueness
of
solutions
to
the
Beltrami
equation
that
Γ
=
f
◦
Γ
◦
f
−1
μ
μ
def
forms
a
group
of
Möbius
transformations
on
C.
Moreover,
from
the
reflection
symmetry
of
μ
,
it
follows
that
f
preserves
the
real
axis,
and
hence
so
does
Γ.
It
thus
follows
that
μ
H/Γ
is
a
Riemann
surface
of
finite
type,
and,
by
the
definition
of
μ,
that
H/Γ
∼
=
X.
This
completes
the
proof.
It
turns
out
that
it
is
this
approach
of
uniformizing
a
single
Riemann
surface
(for
each
g,
r)
and
then
“parallel
transporting”
the
result
over
the
rest
of
the
moduli
space
that
will
carry
over
to
the
p-adic
case.
Uniformization
of
Moduli
Stacks
of
Hyperbolic
Riemann
Surfaces
Let
M
g,r
be
the
moduli
stack
of
r-pointed
smooth
algebraic
curves
of
genus
g
over
C.
Let
M
g,r
be
its
universal
covering
space.
Then
the
problem
of
uniformization
of
moduli
is
g,r
.
From
the
point
of
view
of
the
Beltrami
equation,
to
give
an
explicit
representation
of
M
this
amounts
to
finding
a
small,
finite-dimensional
subspace
T
of
the
space
of
Beltrami
differentials
μ
such
that
the
assignment
μ
→
X
μ
defines
a
covering
space
map
T
→
M
g,r
.
We
begin
by
fixing
a
“base
point”
of
M
g,r
,
which
corresponds
to
a
hyperbolic
Riemann
surface
X.
Let
M(X)
be
the
space
of
Beltrami
differentials
on
X.
Let
Q
be
the
space
of
holomorphic
quadratic
differentials
on
X
with
at
most
simple
poles
at
the
punctured
points.
Then
there
are
two
approaches
to
defining
morphisms
from
open
subsets
of
Q
into
spaces
of
Beltrami
differentials.
The
first
approach
is
that
of
Teichmüller.
In
this
approach,
if
φ
∈
Q,
we
define
a
norm
def
φ
=
|φ|
X
Let
V
⊆
Q
be
the
set
of
φ
with
φ
≤
1.
Then
Teichmüller’s
uniformization
map,
for
(nonzero)
φ
∈
V
,
is
given
by
14
def
φ
→
μ
φ
=
(φ)
φ
|φ|
where
φ
∈
Γ(X,
ω
⊗2
X
)
is
the
complex
conjugate
of
φ.
It
is
easy
to
see
that
μ
φ
defines
a
Beltrami
differential
on
X.
Thus,
we
get
a
morphism
V
→
M(X).
If
we
compose
φ
→
μ
φ
with
μ
→
X
μ
,
we
get
a
morphism
V
→
M
g,r
The
main
result
of
Teichmüller
theory
(see,
e.g.,
[Gard],
Chapter
6)
is
that
this
morphism
g,r
.
One
advantage
of
this
approach
is
that
it
admits
induces
an
isomorphism
of
V
onto
M
a
very
satisfying
geometric
interpretation
in
terms
of
a
foliation
on
X
induced
by
φ
and
deforming
X
into
X
μ
φ
by
deforming
a
canonical
coordinate
arising
from
the
foliation.
The
main
disadvantage
of
this
approach
from
our
point
of
view,
however,
is
that
the
morphism
φ
→
μ
φ
is
neither
holomorphic
nor
anti-holomorphic.
Thus,
it
seems
hopeless
to
try
to
find
an
algebraic
version
of
Teichmüller’s
map.
On
the
other
hand,
Bers’
approach
is
as
follows.
Since
we
now
know
that
X
can
be
uniformized
by
the
upper
half
plane,
let
v
X
be
the
hyperbolic
volume
element
on
X
induced
by
the
hyperbolic
volume
element
v
H
=
dx
∧
dy
y
2
on
the
upper
half
plane.
Let
X
c
be
the
conjugate
Riemann
surface
to
X.
That
is,
the
underlying
topological
manifold
of
X
c
is
the
same
as
that
of
X,
but
the
holomorphic
functions
on
X
c
are
exactly
the
anti-holomorphic
functions
on
X.
Suppose
that
φ
∈
Q.
Then
by
conjugating
the
“input
variable,”
we
obtain
that
φ
defines
a
section
φ
c
of
ω
⊗2
X
c
.
Now
define
def
−2φ
μ
φ
=
c
v
X
c
Then
for
some
appropriate
(see
[Gard],
pp.
100-104)
open
set
V
⊆
Q,
this
μ
φ
defines
a
Beltrami
differential
on
X
c
.
Integrating,
we
get
a
Riemann
surface
X
μ
c
.
Then
the
assignment
φ
→
X
μ
c
defines
a
morphism
V
→
M
c
g,r
where
the
superscript
“c”
denotes
the
conjugate
complex
manifold.
This
morphism
induces
c
g,r
([Gard],
p.
101).
The
important
thing
here
is
that
the
an
isomorphism
of
V
onto
M
correspondence
φ
→
μ
φ
is
holomorphic.
Since
μ
→
X
μ
c
is
always
holomorphic,
it
thus
15
c
is
biholomorphic.
Put
another
way,
we
have
a
follows
that
the
isomorphism
V
∼
=
M
g,r
holomorphic
embedding
g,r
→
Q
c
B
:
M
which
is
called
the
Bers
embedding.
This
embedding
will
be
central
to
our
entire
treatment
of
the
complex
theory,
and
its
p-adic
analogue
will
be
central
to
our
treatment
of
the
p-adic
theory.
Quasidisks
and
the
Bers
Embedding
One
can
also
define
the
Bers
embedding
in
terms
of
Bers’
simultaneous
uniformization
and
Schwarzian
derivatives.
For
details,
see
[Gard],
pp.
100-101.
To
do
this,
we
fix
an
with
H.
Let
H
c
be
the
lower
half
plane.
Thus,
if
H
uniformizes
X,
isomorphism
of
X
c
then
H
naturally
uniformizes
X
c
.
Let
Γ
be
the
group
of
Möbius
transformiations
of
C
→
X.
Then
we
may
think
of
the
space
which
are
the
covering
transformations
for
H
=
X
M(X
c
)
of
Beltrami
differentials
on
X
c
as
the
space
of
Beltrami
differentials
on
H
c
which
are
invariant
under
Γ.
Let
μ
∈
M(X
c
).
Let
f
μ
:
C
→
C
be
the
unique
quasiconformal
homeomorphism
which
fixes
0
and
1,
goes
to
infinity
at
infinity,
has
Beltrami
coefficient
μ
on
H
c
and
is
conformal
on
H.
Let
Γ
μ
=
f
μ
◦Γ◦(f
μ
)
−1
.
Then
it
follows
from
the
uniqueness
of
solutions
to
the
Beltrami
equation
that
Γ
μ
forms
a
group
of
Möbius
transformations
of
C.
Moreover,
we
have
conformal
isomorphisms
f
μ
(H
c
)/Γ
μ
∼
=
X
μ
c
;
f
μ
(H)/Γ
μ
∼
=
X
It
follows
that
if
we
take
the
Schwarzian
derivative
of
the
conformal
“quasidisk”
embedding
f
μ
|
H
:
H
→
C
we
get
a
Γ-invariant
quadratic
differential
on
H,
hence
a
quadratic
differential
φ
(with
at
most
simple
poles
at
the
punctures)
on
X.
The
content
of
the
Lemma
of
Ahlfors-Weill
([Gard],
p.
100)
is
that
the
assignment:
X
μ
c
→
φ
c
→
Q.
On
the
one
hand,
this
description
of
the
Bers
embedding
is
is
equal
to
B
c
:
M
g,r
geometrically
more
satisfying
than
the
definition
given
in
the
previous
subsection,
but
it
has
the
disadvantage
that
it
obscures
the
relationship
between
the
hyperbolic
and
parabolic
cases.
So
far
we
have
been
mainly
discussing
the
hyperbolic
case,
but
we
shall
discuss
the
parabolic
case
later.
16
The
Infinitesimal
Form
of
the
Modular
Uniformizations
Often
it
is
useful
to
express
these
modular
uniformizations
in
their
infinitesimal
form,
as
metrics.
On
the
one
hand,
the
global
uniformizations
can
always
be
essentially
recovered
by
integrating
the
metrics,
and
on
the
other
hand,
metrics,
being
local
in
nature,
can
often
be
studied
more
easily.
In
the
Teichmüller
case,
if
K
is
defined
by
φ
=
K
−
1
K
+1
g,r
,
given
by
then
one
obtains
a
distance
function
on
M
d(X,
X
μ
φ
)
=
1
log(K)
2
which
turns
out
to
be
equal
to
the
general
hyperbolic
distance
introduced
by
Kobayashi
for
an
arbitrary
hyperbolic
complex
manifold
(see
[Gard],
Chapter
7,
for
an
exposition).
The
infinitesimal
form
of
this
distance
is
given
by
the
norm
φ
=
X
|φ|
on
quadratic
differentials
(see
[Royd]).
We
shall
be
more
interested
in
the
case
of
the
Bers
embedding
g,r
→
Q
c
B
:
M
By
using
the
hyperbolic
volume
form
v
X
on
X,
we
obtain
the
Weil-Petersson
inner
product:
def
φ,
ψ
=
X
φ
·
ψ
v
X
for
φ,
ψ
∈
Q.
It
is
a
result
of
Weil
and
Ahlfors
that
the
resulting
metric,
called
the
Weil-
Petersson
metric
on
M
g,r
,
is
Kähler.
Moreover,
if
we
differentiate
B,
we
get,
at
X,
a
map
on
tangent
spaces
dB
:
Q
∨
→
Q
c
whose
inverse
is
exactly
the
morphism
Q
c
→
Q
∨
defined
by
the
Weil-Petersson
inner
prod-
uct.
Finally,
the
coordinates
obtained
from
the
Bers
embedding
are
canonical
coordinates
for
the
Weil-Petersson
metric
([Royd]).
(We
shall
review
the
general
theory
of
canonical
coordinates
associated
to
a
real
analytic
Kähler
metric
in
§2.)
It
turns
out
that
it
is
precisely
the
p-adic
analogue
of
the
Weil-Petersson
metric
that
will
play
a
central
role
in
this
paper.
17
Coordinates
of
Degeneration
While
the
Bers
coordinates
are
useful
for
understanding
what
happens
in
the
interior
of
M
g,r
,
they
are
not
so
useful
for
understanding
what
happens
as
one
goes
out
to
the
boundary,
that
is,
as
the
Riemann
surface
degenerates
to
a
Riemann
surface
with
nodes.
To
study
this
sort
of
degeneration,
one
fixes
a
decomposition
of
the
Riemann
surface
into
“pants,”
which
are
topologically
equivalent
to
an
open
disk
with
two
smaller
disks
in
the
interior
removed.
For
a
detailed
description
of
the
theory
of
pants
and
the
coordinates
they
define,
we
refer
to
[Abikoff],
Chapter
2.
In
summary,
what
happens
is
the
following.
Let
X
be
a
hyperbolic
Riemann
surface
(of
genus
g
with
r
punctures),
with
a
decomposition
into
pants.
We
shall
call
the
curves
on
X
which
occur
in
the
boundary
of
the
pants
partition
curves.
There
are
exactly
3g
−
3
+
r
partition
curves,
α
1
,
.
.
.
,
α
3g−3+r
.
We
assume
that
this
decomposition
is
“maximal”
in
the
sense
that
each
partition
curve
is
a
simple
closed
geodesic
(in
the
hyperbolic
metric
on
X).
Then
it
turns
out
that
the
isomorphism
class
of
X
as
a
Riemann
surface
is
completely
determined
by
3g
−
3
+
r
complex
numbers
ζ
i
=
l
i
e
iθ
(i
=
1,
.
.
.
,
3g
−
3
+
r),
one
for
each
partition
curve.
Basically
l
i
describes
the
circumference
of
the
partition
curve
α
i
,
while
θ
i
describes
the
angle
of
twisting
involved
in
gluing
together
the
boundary
curves
of
two
neighboring
pants
to
form
α
i
.
These
coordinates
ζ
i
are
called
the
Fenchel-Nielsen
coordinates
of
X.
The
degeneration
corresponding
to
pinching
α
i
to
a
node
is
given
by
l
i
→
0.
This
degeneration
respects
the
hyperbolic
metrics
involved:
that
is,
if
a
family
of
smooth
X
t
degenerates
to
a
nodal
Riemann
surface
Z,
then
the
hyperbolic
metrics
on
the
X
t
degenerate
to
the
hyperbolic
metric
on
Z
(given
by
taking
the
hyperbolic
metric
on
the
smooth
subsurface
of
Z
which
is
the
complement
of
the
nodes).
Thus,
the
Fenchel-Nielsen
coordinates
have
the
virtue
of
admitting
a
very
satisfying
differential-geometric
description
(as
just
summarized),
but
the
disadvantage
of
not
being
holomorphic.
On
the
other
hand,
one
can
define
holomorphic
coordinates
(as
in
[Wolp]),
as
follows.
Recall
the
quasidisk
description
of
the
Bers
embedding.
Thus,
we
had
a
μ
∈
M(X
c
),
and
a
quasiconformal
homeomorphism
f
μ
:
C
→
C,
together
with
a
new
group
of
Möbius
transformations
Γ
μ
.
Then
each
α
i
defines
(by
integration)
an
element
A
i
∈
Γ
μ
.
Up
to
conjugation,
A
i
is
of
the
form
z
→
m
i
·
z
for
some
m
i
∈
C
with
|m
i
|
>
1.
This
complex
number
m
i
is
uniquely
defined.
Then
the
coordinates
X
μ
c
→
(m
1
,
.
.
.
,
m
3g−3+r
)
are
holomorphic
in
μ.
In
[Wolp],
the
relationship
between
these
coordinates
and
the
Bers
coordinates
is
studied.
In
these
coordinates,
the
degeneration
of
X
μ
c
corresponding
to
the
case
where
the
partition
curve
α
i
is
pinched
to
a
node
is
given
by
m
i
→
1.
It
turns
out
that
these
coordinates
are
probably
the
best
complex
analogue
to
the
“multiplicative
parameters
at
infinity”
that
we
construct
in
the
p-adic
case.
18
The
Parabolic
Case
So
far
we
have
mainly
been
discussing
the
case
of
hyperbolic
Riemann
surfaces,
since
this
case
is
by
far
the
most
interesting.
However,
often
it
is
very
difficult
to
make
explicit
computations
for
hyperbolic
Riemann
surfaces.
Thus,
in
order
to
get
one’s
bearings,
it
is
sometimes
useful
to
consider
the
analogous
constructions
in
the
parabolic
case,
where
explicit
computations
are
much
easier
to
carry
out.
Let
X
be
a
parabolic
Riemann
surface.
Then
X
is
either
compact
of
genus
1,
or
it
is
isomorphic
to
the
projective
line
minus
two
points.
We
shall
mainly
be
interested
in
the
compact
case,
where
there
are
nontrivial
moduli.
Thus,
let
X
be
compact
of
genus
1.
Then
one
can
carry
out
Teichmüller
theory
in
this
case
(as
in
[Lehto],
Chapter
V,
§6).
One
can
also
define
a
parabolic
analogue
of
the
Bers
embedding,
as
follows.
Namely,
we
simply
copy
the
formula
def
−2φ
μ
φ
=
c
v
X
c
of
the
hyperbolic
case,
except
that
we
take
v
X
c
to
be
the
parabolic
volume
element
(as
opposed
to
the
hyperbolic
volume
element)
on
X
c
,
with
X
c
v
X
c
=
1.
Then
one
sees
(as
in
[Lehto],
p.
220)
that
one
obtains
a
holomorphic
embedding
1,0
→
Q
c
B
1,0
:
M
whose
image
is
an
open
disk
D
⊆
Q
c
of
some
radius.
One
can
also
define
a
Weil-Petersson
metric
on
M
1,0
by
simply
replacing
the
hyperbolic
volume
element
used
before
by
the
parabolic
volume
element.
A
simple
calculation
then
reveals
that
one
obtains
the
standard
hyperbolic
metric
on
the
open
disk
D.
In
particular,
(just
as
in
the
hyperbolic
case),
the
standard
coordinate
on
D
is
normal
at
0
for
the
Weil-Petersson
metric.
One
thing
that
is
interesting
about
this
parabolic
case
is
that
even
though
the
complex
analytic
stacks
M
1,0
and
M
1,1
are
isomorphic,
the
“Bers
theory”
differs
substantially
in
1,1
is
far
from
being
an
open
disk.
the
two
cases.
For
instance,
the
Bers
embedding
of
M
In
fact,
(as
the
author
was
told
by
C.
McMullen)
the
boundary
of
this
hyperbolic
Bers
embedding
has
lots
of
cusps.
A
computer-generated
illustration
of
this
boundary
appears
in
[McM].
Also,
it
is
not
difficult
to
show
that
the
Weil-Petersson
metrics
are
quite
different.
This
contrasts
considerably
with
the
“Teichmüller
theory”
of
M
1,0
and
M
1,1
:
Indeed,
since
Teichmüller’s
metric
always
coincides
with
Kobayashi’s
intrinsic
hyperbolic
metric,
it
follows
that
the
Teichmüller
metrics
of
M
1,0
and
M
1,1
coincide.
Real
Curves
A
Riemann
surface
X
of
finite
type
is
called
real
if
X
∼
=
X
c
.
In
other
words,
this
means
that
the
C-valued
point
defined
by
X
in
the
algebraic
stack
(M
g,r
)
R
(over
Spec(R))
is,
19
in
fact,
defined
over
R
(up
to
perhaps
reordering
the
marked
points).
Various
interesting
properties
of
real
Riemann
surfaces
(related
to
uniformization
theory)
are
studied
in
[Falt2].
Many
of
these
properties
are
obtained
by
looking
at
various
one-dimensional
real
analytic
submanifolds
of
a
real
X.
From
our
point
of
view,
however,
the
notable
fact
about
real
hyperbolic
Riemann
surfaces
X
is
the
following.
Let
φ
:
X
∼
=
X
c
be
a
holomorphic
isomorphism.
For
simplicity,
suppose
that
there
exists
a
point
x
∈
X
such
that
φ(x)
=
x
c
,
and
that
φ
c
◦
φ
=
id
X
.
Fix
∼
c
∼
an
isomorphism
X
=
H
c
.
On
the
other
hand,
φ
=
H.
This
induces
an
isomorphism
X
c
induces
a
holomorphic
isomorphism
φ
:
H
→
H
.
Let
C
:
H
c
→
H
be
the
conjugation
Thus,
ψ
is
an
anti-holomorphic
automorphism
of
H.
Now
let
map.
Let
ψ
=
C
◦
φ.
Π
C
=
π
1
(X,
x).
Since
X
c
has
the
same
underlying
topological
space
as
X,
we
have
Π
C
=
π
1
(X
c
,
x
c
).
Thus,
φ
induces
an
automorphism
φ
Π
of
Π
C
of
degree
2.
Let
Π
R
be
the
extension
1
→
Π
C
→
Π
R
→
Gal(C/R)
→
1
which
is
the
crossed
product
of
Π
C
with
Gal(C/R)
given
by
letting
the
nontrivial
element
of
Gal(C/R)
act
on
Π
C
by
means
of
φ
Π
.
Now
let
us
consider
the
Lie
group
def
G(R)
=
{M
∈
GL
2
(R)|
det(M
)
=
±1}/{±1}
Thus,
PSL
2
(R)
⊆
G(R)
⊆
GL
±
2
(R),
so
we
can
write
ρ
C
:
Π
C
→
GL
±
2
(R)
for
the
canonical
representation
of
X
(uniformized
by
the
upper
half
plane
H).
Note
that
a
b
the
full
group
GL
±
∈
GL
±
2
(R)
acts
on
the
upper
half
plane
as
follows:
if
A
=
2
(R),
c
d
we
let
A(z)
=
aw
+
b
cw
+
d
where
w
=
z
(respectively,
w
=
z)
if
det(A)
is
positive
(respectively,
negative).
Thus,
the
map
defined
by
A
is
a
holomorphic
(respectively,
anti-holomorphic)
automorphism
of
H
if
det(A)
is
positive
(respectively,
negative).
In
particular,
the
anti-holomorphic
automorphism
ψ
:
H
→
H
defines
an
element
(which
by
abuse
of
notation
we
call)
ψ
∈
G(R).
Now
note
that
if
γ
∈
Π
C
,
then
ψ
·
ρ(γ)
·
ψ
−1
=
ρ(φ
Π
(γ)).
Thus,
by
mapping
the
nontrivial
element
of
Gal(C/R)
in
the
crossed
product
definition
of
Π
R
to
ψ,
we
see
that
we
obtain
a
natural
homomorphism
ρ
R
:
Π
R
→
GL
±
2
(R)
20
which
extends
ρ
C
and
is
such
that
the
composite
with
the
determinant
det
:
GL
±
2
(R)
→
R
×
is
trivial
on
Π
C
and
equal
to
the
sign
representation
on
Gal(C/R).
It
is
this
repre-
sentation
ρ
R
that
will
be
relevant
to
our
discussion
of
the
p-adic
case.
§2.
Translation
into
the
p-adic
Case
In
this
Section,
we
discuss
the
dictionary
for
translating
the
complex
analytic
theory
of
§1
into
the
p-adic
results
discussed
in
§0.
Undoubtedly,
the
most
fundamental
tool,
which
is,
in
fact,
of
an
algebraic,
not
an
arithmetic
nature,
is
the
systematic
use
of
the
indigenous
bundles
of
[Gunning].
This
enables
one
to
get
rid
of
the
upper
half
plane,
and
thus
to
bring
uniformization
theory
into
a
somewhat
more
algebraic
setting.
In
any
sort
of
nontrivial
arithmetic
theory
of
this
nature,
however,
algebraic
manipulations
alone
can
never
be
enough.
Thus,
the
fundamental
arithmetic
observation
is
the
following:
Kähler
metrics
in
the
complex
case
correspond
to
Frobenius
actions
in
the
p-adic
case.
Since
one
typically
gets
a
natural
Frobenius
action
for
free
modulo
p,
a
Frobenius
action
typically
means
a
canonical
lifting
of
the
natural
Frobenius
action
modulo
p.
In
fact,
in
some
sense,
if
one
sorts
through
the
complex
analytic
theory
reviewed
§1,
one
can
essentially
distill
everything
down
to
two
objects,
both
of
which
happen
to
be
Kähler
metrics:
(1)
the
hyperbolic
metric
on
a
hyperbolic
Riemann
surface
(which
encodes
the
upper
half
plane
uniformization);
and
(2)
the
Weil-Petersson
metric
on
the
moduli
space
(which
encodes
the
Bers
uniformization).
Moreover,
these
two
metrics
are
related
to
each
other
in
the
sense
that
the
latter
is
essen-
tially
the
push-forward
of
the
former.
In
a
similar
way,
the
p-adic
theory
revolves
around
two
fundamental
Frobenius
liftings:
(1)
the
canonical
Frobenius
lifting
on
a
canonical
hyperbolic
curve;
and
(2)
the
canonical
Frobenius
lifting
on
a
certain
stack
which
is
étale
over
the
moduli
stack.
The
goal
of
this
Section
is
to
explain
this
analogy
in
greater
detail.
21
Gunning’s
Theory
of
Indigenous
Bundles
Let
X
be
a
compact
hyperbolic
Riemann
surface.
Let
H
→
X
be
its
uniformization
by
the
upper
half
plane.
Then
by
considering
the
covering
transformations
of
H
→
X,
we
get
a
homomorphism
(unique
up
to
conjugation)
ρ
:
π
1
(X)
→
Aut(H)
⊆
PSL
2
(R)
which
we
call
the
canonical
representation
of
X.
If
we
regard
ρ
as
defining
a
morphism
into
PSL
2
(C),
then
we
obtain
(in
the
usual
fashion),
a
local
system
of
P
1
-bundles
on
X,
which
thus
gives
us
a
holomorphic
P
1
-bundle
with
connection
(P,
∇
P
)
on
X.
By
Serre’s
GAGA,
(P,
∇
P
)
is
necessarily
algebraic.
It
turns
out
that
P
is
always
isomorphic
to
a
certain
P
1
-bundle
of
jets
(which
is
also
entirely
algebraic).
Thus,
the
upper
half
plane
uniformization
may
be
thought
of
as
just
being
a
special
choice
of
connection
∇
P
.
A
pair
“like”
(P,
∇
P
)
(satisfying
certain
technical
properties
discussed
in
Chapter
I,
§2)
is
called
an
indigenous
bundle.
By
working
with
log
structures,
one
can
also
define
indigenous
bundles
in
a
natural
way
for
smooth
X
with
punctures,
as
well
as
for
nodal
X.
As
emphasized
earlier,
the
point
of
dealing
with
indigenous
bundles
is
that
they
allow
one
to
translate
the
upper
half
plane
uniformization
into
the
purely
algebraic
information
of
a
connection
on
P
.
Of
course,
how
one
chooses
this
particular
special
connection
on
P
is
very
nontrivial
arithmetic
issue.
We
shall
call
the
pair
(P,
∇
P
)
consisting
of
P
equipped
with
this
particular
connection
the
canonical
indigenous
bundle
on
X.
Universally,
over
the
moduli
stack
M
g,r
(of
stable
r-pointed
curves
of
genus
g
over
C),
the
space
of
all
indigenous
bundles
forms
a
holomorphic
torsor
S
g,r
→
M
g,r
over
the
logarithmic
cotangent
bundle
Ω
log
M
g,r
/C
of
M
g,r
.
In
the
holomorphic
category,
we
shall
see
(in
Chapter
I,
§3)
that
this
torsor
is
highly
nontrivial.
In
the
real
analytic
category,
however,
the
canonical
indigenous
bundle
determines
a
trivializing
section
s
H
:
S
g,r
→
M
g,r
of
this
torsor.
In
fact,
indigenous
bundles
also
allow
us
to
translate
such
differential-geometric
in-
formation
as
the
hyperbolic
geometry
of
X
into
algebraic
terms.
For
instance,
consider
the
degeneration
of
Riemann
surfaces
from
the
point
of
view
of
hyperbolic
geometry.
As
reviewed
in
§1,
this
may
be
thought
of
in
terms
of
certain
geodesic
partition
curves
whose
lengths
go
to
zero
as
a
family
of
smooth
X
t
degenerates
to
a
nodal
Riemann
surface
Z.
From
the
complex
theory,
we
know
that
the
hyperbolic
metric
on
X
t
degenerates
to
the
hyperbolic
metric
on
Z.
Using
indigenous
bundles,
we
can
translate
this
into
a
more
al-
gebraic
statement
as
follows:
We
define
the
canonical
indigenous
bundle
on
Z
to
be
the
22
indigenous
bundle
obtained
by
gluing
together
the
canonical
indigenous
bundles
of
the
pointed
Riemann
surfaces
occurring
in
the
normalization
of
Z.
Then
the
statement
is
that
as
X
t
degenerates
to
Z,
the
canonical
indigenous
bundle
on
X
t
degenerates
to
the
canon-
ical
indigenous
bundle
on
Z.
The
statement
that
the
lengths
of
the
partition
geodesics
go
to
zero
then
takes
the
form
that
the
monodromy
of
the
limit
indigenous
bundle
of
the
canonical
indigenous
bundles
of
the
X
t
’s
is
nilpotent
at
the
nodes.
The
Canonical
Coordinates
Associated
to
a
Kähler
Metric
In
this
subsection
we
discuss
how
a
Kähler
metric
on
a
complex
manifold
can
be
used
to
define
canonical
affine,
holomorphic
coordinates
on
the
manifold
locally
in
a
neighborhood
of
a
given
point.
We
believe
that
what
is
discussed
here
is
well-known,
but
our
point
of
view
is
somewhat
different
from
that
usually
taken
in
the
literature.
Let
M
be
a
smooth
complex
manifold
of
complex
dimension
m.
The
complex
analytic
structure
on
M
defines,
in
particular,
a
real
analytic
structure
on
M
.
Let
μ
be
a
real
analytic
(1,
1)-form
on
M
that
defines
a
Kähler
metric
on
M
.
In
particular,
μ
is
a
closed
differential
form.
Let
M
c
be
the
conjugate
complex
manifold
to
M
:
that
is
to
say,
we
take
M
c
to
be
that
complex
manifold
which
has
the
same
underlying
real
analytic
manifold
structure
as
M
,
but
whose
holomorphic
functions
are
the
anti-holomorphic
functions
of
M
.
Let
us
fix
a
point
e
∈
M
.
Let
N
be
the
germ
of
a
complex
manifold
obtained
by
localizing
the
complex
manifold
M
c
×
M
at
(e,
e)
∈
M
c
×
M
(where
this
last
expression
makes
sense
since
M
c
has
the
same
underlying
set
as
M
).
Let
Ω
hol
(respectively,
Ω
ant
)
be
the
holomorphic
vector
bundle
on
N
obtained
by
pulling
back
the
bundle
Ω
M
(respectively,
Ω
M
c
)
of
holomorphic
differentials
on
M
(respectively,
M
c
)
to
M
c
×
M
via
the
projection
M
c
×
M
→
M
(respectively,
M
c
×
M
→
M
c
),
and
then
restricting
to
N
.
Thus,
in
summary,
we
have
a
2m-dimensional
germ
of
a
complex
manifold
N
,
together
with
two
m-dimensional
holomorphic
vector
bundles
(locally
free
sheaves)
Ω
hol
and
Ω
ant
on
N
.
Note
that
locally
at
e
∈
M
,
the
fact
that
μ
is
real
analytic
means
that
we
can
write
μ
as
a
convergent
power
series
in
holomorphic
and
anti-holomorphic
local
coordinates
at
e.
In
other
words,
if
we
restrict
μ
to
N
,
we
may
regard
μ|
N
as
defining
a
holomorphic
section
of
Ω
hol
⊗
O
N
Ω
ant
(where
O
N
is
the
sheaf
of
holomorphic
functions
on
N
).
Let
d
hol
(respectively,
d
ant
)
be
the
exterior
derivative
on
N
with
respect
to
the
variables
coming
from
M
(respectively,
M
c
).
Note
that
since
Ω
hol
is
constructed
via
pull-back
from
M
,
we
can
apply
d
ant
to
sections
of
Ω
hol
.
We
thus
obtain
a
sort
of
de
Rham
complex
with
respect
to
d
ant
:
0
−→
Ω
hol
d
ant
−→
Ω
hol
⊗
O
N
Ω
ant
d
ant
−→
Ω
hol
⊗
O
N
(∧
2
Ω
ant
)
d
ant
−→
...
Relative
to
this
complex,
the
section
μ|
N
of
Ω
hol
⊗
Ω
ant
satisfies
d
ant
μ|
N
=
0
(since
μ
is
a
closed
form).
It
thus
follows
from
the
Poincaré
Lemma
that
there
exists
a
(holomorphic)
section
α
of
Ω
hol
that
vanishes
at
(e,
e)
∈
N
and
satisfies
d
ant
α
=
μ|
N
.
Let
M
e
be
the
germ
of
a
complex
manifold
obtained
by
localizing
M
at
e
∈
M
.
Let
23
ι
:
M
e
c
→
N
be
the
inclusion
induced
by
the
map
M
c
→
M
c
×M
that
takes
f
∈
M
c
to
(f,
e)
∈
M
c
×M
.
Then
ι
∗
(α)
defines
a
holomorphic
morphism
β
:
M
e
c
→
Ω
M,e
,
where
Ω
M,e
is
the
affine
complex
analytic
space
defined
by
the
cotangent
space
of
M
at
e.
Note,
moreover,
that
although
α
(as
chosen
above)
is
not
unique,
β
is
nonetheless
independent
of
the
choice
of
α.
Moreover,
β
is
an
immersion:
Indeed,
to
see
this,
it
suffices
to
check
that
the
map
induced
by
β
on
tangent
spaces
is
an
isomorphism,
but
this
follows
from
the
fact
that
d
ant
α
=
μ|
N
,
and
the
fact
that
the
Hermitian
form
defined
by
μ
is
nondegenerate.
In
summary,
we
see
that
from
the
Kähler
metric
μ,
we
obtain
a
canonical
holomorphic
local
affine
uniformization
β
c
:
M
e
→
Ω
c
M,e
Pulling
back
the
standard
affine
coordinates
on
Ω
c
M,e
gives
us
a
canonical
collection
of
holomorphic
coordinates
on
M
e
.
Definition
2.1.
We
shall
refer
to
these
coordinates
as
the
canonical
holomorphic
local
coordinates
of
the
Kähler
manifold
(M,
μ)
at
e.
We
shall
refer
to
β
c
as
the
canonical
local
affine
uniformization
of
the
Kähler
manifold
(M,
μ)
at
e.
Now
let
us
consider
some
basic
well-known
examples:
Example
1.
Let
M
=
{z
∈
C|
|z|
<
1},
with
the
standard
hyperbolic
metric
√
2dz∧dz
.
1−(z·z)
hol
Then
z
is
a
canonical
coordinate
at
0.
Indeed,
to
see
this
it
suffices
to
note
that
d
(z·dz)
=
dz
∧
dz,
which
is
equal
to
the
metric
modulo
the
ideal
generated
by
z
in
O
N
.
Note
that
by
the
Köbe
uniformization
theorem,
this
example
essentially
covers
all
hyperbolic
Riemann
surfaces.
Example
2.
Let
M
be
the
Teichmüller
space
of
Riemann
surfaces
of
genus
g
with
r
punctures,
where
2g
−
2
+
r
≥
1.
Then
as
stated
earlier,
it
is
known
([Royd])
that
the
coordinates
arising
from
the
Bers
embedding
are
canonical
coordinates
with
respect
to
the
Weil-Petersson
metric
on
M
.
In
fact,
in
this
case,
by
Theorem
2.3
(proven
below)
the
real
analytic
section
s
H
defined
by
the
canonical
indigenous
bundle
essentially
already
serves
as
an
“α”
in
the
above
discussion.
Thus,
in
a
very
real
sense,
the
section
s
H
already
is
the
Bers
embedding.
24
The
Weil-Petersson
Metric
from
the
Point
of
View
of
Indigenous
Bundles
Let
X
be
a
compact
hyperbolic
Riemann
surface.
Let
(π
:
P
→
X,
∇
P
)
be
the
canonical
indigenous
bundle
on
X.
Let
Ad(P
)
=
π
∗
τ
P/X
be
the
push-forward
of
the
relative
tangent
bundle
of
π.
Thus,
Ad(P
)
is
a
rank
3
vector
bundle
on
X,
equipped
with
a
simple
Lie
algebra
structure,
hence
with
a
nondegenerate
Killing
form
<
−,
−
>:
Ad(P
)
⊗
O
X
Ad(P
)
→
O
X
.
Moreover,
∇
P
induces
a
connection
∇
Ad
on
Ad(P
).
Moreover,
as
an
indigenous
bundle,
Ad(P
)
comes
equipped
with
a
section
σ
:
X
→
P
(the
“Hodge
section”)
which
defines
a
Hodge
filtration
F
·
(Ad(P
))
on
Ad(P
).
(See
Chapter
I
for
more
1
(Ad(P
),
∇
Ad
)
module
details.)
At
any
rate,
we
can
take
the
first
de
Rham
cohomology
H
DR
of
(Ad(P
),
∇
Ad
).
The
Hodge
filtration
on
Ad(P
)
then
defines
a
Hodge
filtration
on
the
de
Rham
cohomology,
hence
an
exact
sequence:
⊗2
1
0
→
H
0
(X,
ω
X
)
→
H
DR
(Ad(P
),
∇
Ad
)
→
H
1
(X,
τ
X
)
→
0
On
the
other
hand,
recall
the
representation
that
we
used
to
define
(P,
∇
P
):
ρ
:
π
1
(X)
→
Aut(H)
⊆
PSL
2
(R)
Let
Ad(V
R
)
denote
the
π
1
(X)-module
obtained
by
letting
π
1
(X)
act
on
the
Lie
algebra
def
sl
2
(R)
by
applying
ρ
and
then
conjugating
matrices.
Let
Ad(V
C
)
=
Ad(V
R
)
⊗
R
C.
Then
(it
is
elementary
that)
we
have
a
“comparison
theorem”
that
gives
a
natural
isomorphism
between
the
de
Rham
cohomology
module
just
considered
and
the
group
cohomology
of
Ad(V
C
):
1
H
DR
(Ad(P
),
∇
Ad
)
∼
=
H
1
(π
1
(X),
Ad(V
C
))
On
the
other
hand,
we
also
have:
H
1
(π
1
(X),
Ad(V
C
))
∼
=
H
1
(π
1
(X),
Ad(V
R
))
⊗
R
C
which,
combined
with
the
above
comparison
theorem,
thus
gives
a
real
structure
on
1
H
DR
(Ad(P
),
∇
Ad
).
One
way
to
express
this
real
structure
is
as
an
R-linear
conjugation
1
1
(Ad(P
),
∇
Ad
)
→
H
DR
(Ad(P
),
∇
Ad
).
morphism
(read:
“Frobenius
action”)
c
DR
:
H
DR
Now
let
us
consider
the
relationship
between
c
DR
and
the
Hodge
filtration.
If
we
⊗2
1
compose
the
natural
inclusion
H
0
(X,
ω
X
)
→
H
DR
(Ad(P
),
∇
Ad
)
with
c
DR
followed
by
the
1
1
natural
projection
H
DR
(Ad(P
),
∇
Ad
)
→
H
(X,
τ
X
),
we
obtain
a
C-bilinear
form
β
:
H
0
(X,
ω
⊗2
)
⊗
C
H
0
(X,
ω
⊗2
)
c
→
C
(where
the
superscript
“c”
stands
for
the
complex
conjugate
C-vector
space).
25
Proposition
2.2.
The
form
β
is
precisely
the
Weil-Petersson
metric
on
quadratic
differ-
entials
defined
in
§1
by
means
of
integration.
In
particular,
β
is
nondegenerate.
Proof.
In
order
to
obtain
β,
we
implicitly
used
the
special
case
of
Serre
duality
given
by
⊗2
).
But
in
the
complex
analytic
context,
the
pairing
that
defines
H
1
(X,
τ
X
)
∼
=
H
0
(X,
ω
X
this
sort
of
duality
is
given
by
integrating
the
product
of
((0,
1)−
and
(1,
0)−)
forms.
The
volume
form
v
X
appears
for
the
sake
of
defining
the
duality
between
ω
X
and
ω
X
.
With
these
remarks,
the
claim
of
the
Lemma
becomes
a
tautology.
Now
let
us
recall
the
real
analytic
section
s
H
:
M
g,r
→
S
g,r
.
Since
S
g,r
→
M
g,r
is
a
holomorphic
torsor,
we
may
form
∂s
H
,
which
gives
a
section
of
Ω
log
M
g,r
/C
log
⊗Ω
M
g,r
/C
.
On
the
other
hand,
the
Weil-Petersson
metric
also
defines
a
section
μ
WP
of
Ω
log
M
g,r
log
⊗
Ω
M
g,r
/C
.
/C
Now
we
have
the
following
result
(stated
in
[ZT],
but
from
a
somewhat
different
point
of
view):
Theorem
2.3.
The
form
∂s
H
is
equal
to
μ
WP
.
Proof.
By
introducing
log
structures,
one
can
handle
the
general
case;
here,
for
simplicity,
we
restrict
our
attention
to
the
case
of
smooth
compact
Riemann
surfaces.
Let
us
consider
⊗2
1
)
→
H
DR
(Ad(P
),
∇
Ad
)
with
c
DR
followed
the
composite
of
the
natural
inclusion
H
0
(X,
ω
X
1
1
by
the
natural
projection
H
DR
(Ad(P
),
∇
Ad
)
→
H
(X,
τ
X
);
this
composite
gives
a
C-linear
morphism:
⊗2
)
→
H
1
(X,
τ
X
)
c
H
0
(X,
ω
X
which
is
invertible
by
Lemma
2.2.
Taking
its
inverse,
and
dualizing,
we
obtain
an
element
⊗2
⊗2
c
δ
∈
H
0
(X,
ω
X
)
⊗
C
H
0
(X,
ω
X
)
On
the
other
hand,
sorting
through
the
definitions,
it
is
a
tautology
in
linear
algebra
that
the
value
of
∂s
H
at
the
point
[X]
∈
M
g
is
given
by
δ.
But,
combining
this
with
Lemma
2.2,
we
see
that
we
have
proven
the
Theorem.
The
important
point
here
is
that
this
Theorem
shows
that:
The
Weil-Petersson
metric,
and
hence
the
Bers
embedding,
is
obtained
precisely
by
considering
the
extent
to
which
“Frobenius”
–
i.e.,
complex
conjugation
–
is
compatible
with
the
canonical
indigenous
bundle
section
s
H
.
26
Stated
in
this
way,
the
classical
complex
theory
becomes
all
the
more
formally
analogous
to
the
p-adic
theory
to
be
discussed
in
this
paper.
The
Philosophy
of
Kähler
Metrics
as
Frobenius
Liftings
Before
going
into
a
detailed
account
of
the
correspondence
between
complex
and
p-
adic
results,
we
pause
to
explain
some
of
the
motivation
for
considering
Kähler
metrics
as
Frobenius
liftings.
Let
S
be
a
smooth
p-adic
formal
scheme
over
Z
p
.
A
Frobenius
lifting
on
S
is
a
morphism
Φ
:
S
→
S
whose
reduction
modulo
p
is
equal
to
the
Frobenius
morphism
in
characteristic
p.
Then
the
main
point
of
the
analogy
is
that
just
as
(real
analytic)
Kähler
metrics
define
canonical
coordinates
(as
discussed
above),
Frobenius
liftings
Φ
:
S
→
S
(that
satisfy
a
certain
technical
condition
called
ordinariness
–
see
Chapter
III,
§1
for
details)
also
define
canonical
coordinates,
as
follows:
The
most
basic
example
of
an
ordinary
Frobenius
lifting
is
the
case
when
S
is
the
p-adic
completion
of
Z
p
[T,
T
−1
]
(where
T
is
an
indeterminate),
and
Φ
−1
(T
)
=
T
p
.
Then
the
theory
of
ordinary
Frobenius
liftings
(Chapter
III,
§1)
states
that
by
means
of
a
certain
“integration”
procedure,
every
ordinary
Frobenius
lifting
on
an
arbitrary
S
becomes
(after
completing
at
a
point
of
S)
isomorphic
to
a
product
of
copies
of
this
basic
example.
This
“integration
procedure”
is
thus
analogous
to
the
integration
procedure
just
reviewed
which
allowed
us
to
construct
canonical
coordinates
associated
to
real
analytic
Kähler
metrics.
The
Dictionary
The
fundamental
“nuts
and
bolts”
of
the
complex
theory
lies
in
the
Beltrami
equa-
tion.
Suppose
that
we
think
of
the
Beltrami
equation
not
as
a
differential
equation
whose
unknown
is
the
quasiconformal
function
f
μ
,
but
instead
as
an
equation
whose
unknown
is
the
conformal
quasidisk
embedding
function
f
μ
|
H
(in
the
discussion
of
quasidisks).
A
quasidisk
embedding
of
the
universal
covering
space
of
a
hyperbolic
Riemann
surface
X
defines
an
indigenous
bundle
(P,
∇
P
)
μ
on
X
in
a
natural
way.
Thus,
from
this
point
of
view,
we
can
think
of
the
Beltrami
equation
as
an
equation
whose
unknown
is
(P,
∇
P
)
μ
.
Moreover,
the
Beltrami
coefficient
μ
defines
the
“shearing”
or
distortion
factor
between
z
and
z.
Thus,
in
summary,
we
may
regard
the
Beltrami
equation
as
an
equation
in
the
unknown
(P,
∇
P
)
μ
in
terms
of
the
distortion
factor
(effected
by
the
quasidisk
embedding
f
μ
|
H
)
between
z
and
its
“Frobenius
conjugate”
z.
On
the
other
hand,
the
“nuts
and
bolts”
of
the
p-adic
theory
lies
in
the
study
of
the
Verschiebung
on
indigenous
bundles,
which
occupies
most
of
Chapter
II.
As
a
function
on
the
indigenous
bundles
of
a
hyperbolic
curve
in
characteristic
p,
the
Verschiebung
–
which
is
essentially
the
determinant
of
the
p-curvature
–
measures
the
distortion
factor
between
applying
Frobenius
to
an
infinitesimal
on
the
curve
and
applying
Frobenius
to
an
infinitesimal
motion
in
the
(“quasidisk”)
uniformization
defined
by
the
indigenous
bundle.
Thus,
for
instance,
when
the
p-curvature
is
nilpotent,
there
is
no
distortion
factor,
and
so
the
indigenous
bundle
provides
the
“right”
uniformization
for
the
curve.
In
this
sense,
we
27
feel
that
there
is
an
analogy
between
the
Beltrami
equation
in
the
complex
theory
and
the
Verschiebung
on
indigenous
bundles
in
the
p-adic
theory.
Relative
to
this
analogy,
the
fundamental
existence
and
uniqueness
theorem
for
solu-
tions
to
the
Beltrami
equation
becomes
the
result
(in
Chapter
II)
that
the
Verschiebung
on
indigenous
bundles
is
finite
and
flat.
Since
in
the
p-adic
case,
its
degree
is
not
one,
we
only
have
uniqueness
up
to
a
finite
number
of
possibilities.
This
is
why
we
get
several
distinct
“quasiconformal
equivalence
classes”
in
the
p-adic
case.
Moreover,
the
important
integral
operator
“T
”
–
i.e.,
the
parametrix
to
∂
–
which
gives
the
first
term
in
the
series
expansion
for
f
μ
may
be
regarded
as
having
its
analogue
in
the
p-adic
theory
in
the
infinitesimal
Verschiebung,
which
plays
an
important
role
throughout
the
paper.
More
obvious
is
the
analogy
between
the
canonical
representation
ρ
C
:
π
1
(X)
→
PSL
2
(R)
of
a
hyperbolic
Riemann
surface
(arising
from
the
upper
half
plane
uniformiza-
tion),
and
the
canonical
representation
ρ
∞
:
Π
∞
→
GL
±
2
(Z
p
)
of
an
ordinary
p-adic
curve
(in
Theorem
0.4).
Of
course
in
the
p-adic
case,
Π
∞
has
a
substantial
arithmetic
part
in
addition
to
its
geometric
part.
Although
generally
in
the
complex
case,
there
is
not
much
of
a
Galois
group
to
work
with,
at
least
for
real
curves,
we
saw
at
the
end
of
§1,
that
one
does
get
a
natural
representation
ρ
R
of
the
full
“arithmetic
fundamental
group”
Π
R
into
GL
±
2
(R).
Moreover,
our
approach
to
constructing
ρ
∞
in
the
p-adic
case
is
very
much
akin
to
Bers’
approach
to
constructing
ρ
C
in
the
complex
case:
Namely,
if
one
traces
through
the
proof
(which
lies
in
Chapters
II
through
V),
one
sees
that
effectively
what
we
are
doing
is
noting
that
the
result
is
true
for
totally
degenerate
curves,
and
then
transporting
this
result
over
the
rest
of
the
moduli
stack
of
ordinary
curves.
Next
let
us
consider
metrics
and
geometry.
As
we
stated
earlier,
in
some
sense,
one
can
summarize
the
entire
complex
theory
by
saying:
We
start
with
the
hyperbolic
(Kähler)
metric
on
a
hyperbolic
curve,
define
the
Weil-Petersson
(Kähler)
metric
on
the
moduli
stack
precisely
so
as
to
be
compatible
with
the
hyperbolic
metric
on
the
curves
being
parametrized;
then
our
holomorphic
uniformizations
–
i.e.,
both
the
upper
half
plane
uniformization
of
the
hyperbolic
curve
and
the
Bers
uniformization
of
the
moduli
stack
–
are
obtained
by
“integrating”
the
respective
metrics.
Similarly,
the
fundamental
result
in
the
p-adic
theory
–
namely,
Theorem
0.1
–
is
a
result
about
the
existence
of
certain
Frobenius
liftings
on
the
universal
hyperbolic
curve
and
its
moduli
stack
which
are
uniquely
characterized
by
the
fact
that
they
are
compatible
with
each
other.
Here
the
compatibility
is
expressed
through
the
tool
of
the
canonical
indigenous
bundle.
Then,
by
“integrating”
these
Frobenius
actions,
we
obtain
canonical
(p-adically
holomorphic)
coordinates
(as
in
ord
Corollary
0.2)
on
C
ord
and
N
g,r
.
This
particular
analogy
lies
at
the
heart
of
this
work.
The
Bers
coordinates
and
the
coordinates
of
Chapter
III,
Theorem
2.4,
are
appropriate
in
the
locus
M
g,r
of
smooth
curves.
For
totally
degenerate
curves,
one
has
multiplicative
parameters
(Chapter
III,
Definition
2.7)
which
we
believe
are
analogous
to
the
holomorphic
coordinates
of
degeneration
of
[Wolp],
reviewed
in
§1.
For
instance,
both
sets
of
parameters
are
holomorphic
and
naturally
indexed
by
the
nodes
of
the
totally
degenerate
curve.
For
elliptic
curves
–
regarded
parabolically
–
one
has,
on
the
one
hand,
the
well-known
theory
of
the
hyperbolic
upper
half
plane,
or
unit
disk,
in
the
complex
case,
and
Serre-
28
Tate
theory
in
the
p-adic
case.
It
is
interesting
to
note
that
at
both
types
of
primes
(complex
and
p-adic),
the
parabolic
theory
may
be
obtained
in
a
very
precise
sense
as
the
parabolic
specializations,
respectively,
of
Bers’
theory
and
of
the
hyperbolic
p-adic
theory
developed
in
this
paper.
In
fact,
this
is
one
of
our
reasons
for
feeling
that
the
canonical
p-adic
coordinates
of
Corollary
0.2
are
the
p-adic
analogue,
not
of
Teichmüller’s
coordinates,
but
of
Bers’:
Namely,
in
addition
to
the
fact
that
Teichmüller’s
coordinates
are
not
holomorphic,
whereas
Bers’
are,
Teichmüller
obtains
the
same
coordinates
for
1-pointed
curves
of
genus
1
and
parabolic
elliptic
curves.
On
the
other
hand,
it
is
well-known
that
Bers’
coordinates
are
very
different
for
1-pointed
curves
of
genus
1
and
parabolic
elliptic
curves,
which
is
consistent
with
the
fact
that
the
canonical
coordinates
of
Corollary
0.2
are
also
very
different
for
1-pointed
curves
of
genus
1
and
parabolic
elliptic
curves.
Loose
Ends
We
close
by
saying
that
although,
as
described
above,
there
are
(what
the
author
believes
to
be)
very
strong
analogies
between
Bers’
complex
theory
and
the
p-adic
theory
presented
here,
the
picture
is
by
no
means
complete.
For
instance,
one
fundamental
fact
in
the
complex
case
is
that
all
r-pointed
smooth
curves
of
genus
g
are
quasiconformally
equivalent,
whereas
in
the
p-adic
case,
the
theory
behaves
as
though
there
are
several
different
quasiconformal
equivalence
classes
that
are
permuted
around
to
each
other
by
a
certain
monodromy
action
in
such
a
way
that
there
seems
to
be
no
one
quasiconformal
equivalence
class
“which
is
better
than
the
others.”
Ideally,
one
would
like
to
have
a
much
more
complete
understanding
of
this
phenomenon.
In
particular,
one
would
like
to
know
precisely
how
many
quasiconformal
equivalence
classes
there
are
(at
least
generically),
as
well
as
a
more
explicit
description
of
the
set
of
such
classes.
Also,
I
still
do
not
understand
what
the
complex,
or
global,
analogue
of
a
“canoni-
cal
p-adic
curve”
is.
For
ordinary
elliptic
curves,
since
Serre-Tate
canonical
liftings
have
complex
multiplication,
one
can
ask
what
the
hyperbolic
analogue
of
having
complex
mul-
tiplication
is.
Since
having
complex
multiplication
for
an
elliptic
curve
means
having
lots
of
isogenies,
it
is
natural
to
ask
if
the
proper
hyperbolic
analogue
is
having
lots
of
cor-
respondences,
which
are
a
sort
of
higher
genus
version
of
isogenies.
If
a
hyperbolic
curve
does
have
a
lot
of
correspondences,
then
one
knows
([Marg])
that
the
image
of
its
canonical
representation
is
arithmetic.
In
Chapter
IV,
we
prove
that
a
canonical
curve
has
lots
of
“pseudo-correspondences,”
but
unfortunately,
at
the
time
of
writing,
I
do
not
see
how
to
make
these
pseudo-correspondences
into
genuine
correspondences,
so
that
one
could
apply
Margulis’
result.
Another
issue
that
arises
in
this
connection
is
the
question
of
whether
one
can
characterize
“hyperbolic
curves
with
complex
multiplication”
–
whatever
the
correct
definition
should
be
for
this
term
–
in
terms
of
the
Bers
coordinates.
Finally,
as
the
title
implies,
the
present
work
deals
exclusively
with
the
case
of
ordinary
curves.
In
a
complete
theory,
one
would
like
to
know
what
happens
when
one
has
a
nilpotent
indigenous
bundle
which
is
not
ordinary.
29
At
any
rate,
in
summary,
with
respect
to
these
three
issues
of
quasiconformal
equiv-
alence
classes,
canonical
curves,
and
non-ordinary
curves,
much
work
remains
to
be
done.
We
hope
to
be
able
to
address
these
issues
in
future
papers.
30
Chapter
I:
Crystalline
Projective
Structures
§0.
Introduction
The
purpose
of
this
Chapter
is
to
study
the
algebraic
analogue
of
projective
structures
on
a
Riemann
surface.
In
particular,
we
prove
many
of
the
analogues
of
results
of
[Gunning]
in
a
purely
algebraic
framework,
often
making
use
of
the
crystalline
site
where
complex
analytically
one
would
restrict
to
a
simply
connected
neighborhood
on
which
one
can
integrate.
Unlike
Gunning,
we
make
systematic
use
of
the
log
structures
of
[Kato],
which
enable
us
to
work
with
a
very
general
sort
of
“log-curve,”
that
is,
we
can
handle
the
case
of
curves
with
marked
points,
as
well
as
singular
nodal
curves
on
an
equal
footing
to
the
smooth
case.
In
§1,
we
discuss
the
notion
of
a
Schwarz
structure,
which
is
the
algebraic
analogue
of
[Gunning]’s
projective
structures.
We
relate
Schwarz
structures
to
projective
bundles
with
connections
as
well
as
to
square
differentials,
and
we
show
that
Schwarz
structures
naturally
give
rise
to
a
Schwarzian
derivative.
(Moreover,
in
the
Appendix
to
this
Chapter,
we
show
that
for
P
1
,
this
abstract
notion
of
a
Schwarzian
derivative
essentially
coincides
with
the
classical
Schwarzian
derivative.)
The
characterizing
feature
of
§1
is
that
everything
takes
place
locally
on
the
curve
in
question.
In
§2,
we
discuss
indigenous
bundles
(the
direct
algebraic
analogue
of
[Gunning]’s
indigenous
bundles).
What
distinguishes
§2
from
§1
is
that
in
§2,
we
work
mainly
over
stable
curves,
and
thus
global
issues
on
the
curve
come
into
play.
In
§2,
we
are
still
working
locally,
however,
on
the
base.
In
§3,
we
perform
various
intersection
theory
calculations
that
allow
us
to
prove
that
in
most
cases,
there
do
exist
any
canonical
indigenous
bundles
on
the
universal
smooth
curve
over
a
moduli
stack.
Thus,
in
§3,
we
are
concerned
with
issues
that
are
global
not
only
on
the
curve,
but
also
on
the
base.
It
should
be
said
that
all
the
material
in
this
Chapter
is,
in
some
sense,
“well-known,”
but
I
do
not
know
of
any
modern
reference
that
does
things
from
this
point
of
view.
In
particular,
all
the
references
that
I
know
of
(with
the
exception
of
[Ih],
which
is
algebraic,
but
somewhat
different
in
point
of
view)
discuss
things
only
in
the
complex
analytic
case,
and
often
work
with
“h
αβ
’s”
(i.e.,
cocycle
classes)
rather
than
with
objects
that
have
an
intrinsic
meaning.
§1.
Schwarz
Structures
In
this
Section,
we
introduce
the
crystalline
analogue
of
what
Gunning
calls
“projective
structures
on
a
Riemann
surface.”
(We
shall
call
them
Schwarz
structures
(after
the
Schwarzian
derivative)
to
distinguish
them
from
the
analytic
notion.)
We
begin
by
letting
S
be
a
connected
noetherian
scheme.
Often,
we
shall
prove
results
about
arbitrary
stable
curves
by
working
on
various
compactified
moduli
stacks.
Thus,
even
if
one
is
ultimately
interested
only
in
smooth
curves,
for
certain
proofs,
we
shall
see
that
it
is
useful
to
develop
31
the
machinery
for
arbitrary
stable
curves.
To
deal
with
singular
curves,
we
shall
use
the
theory
of
log
schemes
of
[Kato].
Thus,
we
assume
that
S
has
a
given
fine
([Kato],
§2)
log
structure,
and
denote
the
resulting
log
scheme
by
S
log
.
Notation
and
Basic
Definitions
Definition
1.1.
Let
f
log
:
U
log
→
S
log
be
a
morphism
of
log
schemes
whose
underlying
morphism
of
schemes
f
:
U
→
S
is
of
finite
type,
flat
and
of
relative
dimension
one.
Then
we
shall
say
that
f
log
is
locally
stable
of
dimension
one
if,
for
every
point
u
∈
U
,
there
exist
étale
morphisms
T
→
S
and
V
→
U
×
S
T
,
together
with
v
∈
V
mapping
to
u
∈
U
such
that
when
we
pull-back
the
log
structure
on
S
(respectively,
U
)
to
T
(respectively,
V
)
to
obtain
log
schemes
V
log
and
T
log
,
one
of
the
following
holds:
(1)
V
→
T
is
smooth,
and
V
log
=
V
×
T
T
log
(where
V
and
T
denote
the
log
schemes
with
trivial
log
structure);
or
(2)
V
→
T
is
smooth,
and
there
exists
a
section
s
:
T
→
V
such
that
if
we
denote
by
V
s
the
log
scheme
defined
by
the
relative
divisor
Im(s)
on
V
,
then
V
log
=
V
s
×
T
T
log
;
or
(3)
let
Y
=
Spec(Z[t]);
X
=
Y
[x,
y]/(xy
−
t)
(where
x,
y,
and
t
are
indeter-
minates)
and
endow
Y
(respectively,
X)
with
the
log
structure
arising
from
the
divisor
t
=
0
(respectively,
xy
=
0),
so
we
get
a
morphism
X
log
→
Y
log
of
log
schemes;
then
there
exists
a
morphism
of
log
schemes
T
log
→
Y
log
,
together
with
a
morphism
ζ
log
:
V
log
→
T
log
×
Y
log
X
log
such
that
the
underlying
scheme
morphism
ζ
of
ζ
log
is
étale,
and
the
log
structure
of
V
log
on
V
is
the
pull-back
via
ζ
of
the
log
structure
on
T
log
×
Y
log
X
log
.
In
case
(1)
(respectively,
(2);
(3)),
we
shall
say
that
f
is
smooth
and
unmarked
(respectively,
marked;
singular)
at
u.
Note
that
if
f
log
:
U
log
→
S
log
is
locally
stable
of
dimension
one,
then
it
is
always
log
smooth
([Kato],
§3).
Also,
note
that
by
étale
descent,
the
images
in
U
of
all
the
sections
s
as
in
Case
(2)
above
form
a
divisor
in
U
which
is
étale
over
S.
We
shall
refer
to
this
divisor
as
the
divisor
of
marked
points
in
U
.
Now
let
us
suppose
that
there
exists
an
odd
prime
p
which
is
nilpotent
on
S.
We
also
suppose
that
we
are
given
a
closed
subscheme
S
0
=
V
(I)
⊆
S,
where
the
sheaf
of
ideals
I
has
a
divided
power
structure
γ.
We
denote
the
log
scheme
S
0
×
S
S
log
(where
S
0
and
S
denote
the
log
schemes
which
are
the
respective
schemes
endowed
with
the
trivial
log
structure)
by
S
0
log
.
Let
f
log
:
U
log
→
S
log
be
locally
stable
of
dimension
one.
Then
we
shall
call
a
section
of
D
Δ
(U
log
×
S
log
U
log
)
(the
PD-envelope
of
the
diagonal,
as
in
[Kato],
32
§5)
a
bianalytic
function
over
U
log
.
Note
that
the
bianalytic
functions
form
a
sheaf,
which
we
denote
O
U
bi
,
on
the
étale
site
of
U
.
Let
O
U
denote
the
sheaf
on
the
étale
site
of
U
given
by
considering
ordinary
functions.
Then
the
two
projections
U
×
S
U
→
U
give
rise
to
injections
i
L
:
O
U
→
O
U
bi
and
i
R
:
O
U
→
O
U
bi
whose
images
we
shall
call
the
left-sided
(respectively,
right-sided)
bianalytic
functions
on
U
log
.
We
shall
also
refer
to
right-sided
bianalytic
functions
as
constant
bianalytic
functions,
or
bianalytic
constants.
We
denote
tensor
products
of
an
O
U
-module
F
over
O
U
with
O
U
bi
via
i
L
(respectively,
i
R
)
by
writing
F
on
the
left
(respectively,
right).
Finally,
we
have
a
multiplication
morphism
μ
:
O
U
bi
→
O
U
.
We
denote
the
ideal
subsheaf
of
O
U
bi
which
is
the
kernel
of
μ
by
J
.
We
shall
say
that
a
bianalytic
function
f
over
some
étale
V
→
U
is
a
bianalytic
uniformizer
on
V
if
f
is,
in
fact,
a
section
of
J
which
generates
the
line
bundle
J
/J
[2]
∼
=
ω
U
log
/S
log
as
an
O
U
-
module.
Let
O
U
bi
be
the
completion
of
O
U
bi
with
respect
to
the
divided
powers
J
[i]
,
and
let
J
⊆
O
U
bi
be
the
closure
of
J
in
O
U
bi
.
We
shall
call
sections
of
O
U
bi
biformal
functions,
and
use
similar
terminology
for
biformal
functions
as
we
do
for
bianalytic
functions.
Occasionally,
we
shall
also
need
to
make
use
of
trianalytic
(respectively,
triformal)
functions,
i.e.,
sections
of
D
Δ
(U
log
×
S
log
U
log
×
S
log
U
log
)
(respectively,
its
completion
with
respect
to
the
divided
powers
of
the
diagonal
ideal).
We
denote
the
sheaf
of
trianalytic
functions
(respectively,
triformal)
on
the
étale
site
of
U
by
O
U
tr
(respectively,
O
U
tr
),
and
we
have
left,
right,
and
middle
injections
j
1
,
j
2
,
j
3
:
O
U
→
O
U
tr
,
as
well
as
injections
j
12
,
j
23
,
j
13
:
O
U
bi
→
O
U
tr
.
We
shall
apply
similar
terminology
and
notation
to
trianalytic
or
triformal
functions
to
that
applied
already
to
bianalytic
functions.
In
particular,
we
shall
call
trianalytic
functions
that
are
in
the
image
of
j
23
trianalytic
constants.
Definition
1.2.
Let
S
⊆
O
U
bi
be
a
subsheaf
in
the
category
of
sets.
We
shall
call
S
a
Schwarz
(respectively,
pre-Schwarz)
structure
on
U
log
if
étale
locally
on
U
(i.e.,
for
some
étale
cover
V
→
U
),
S
has
the
following
form:
there
exists
some
biformal
uniformizer
z
∈
Γ(V,
S)
such
that
for
every
étale
W
→
V
,
and
every
section
f
∈
Γ(W,
O
U
bi
),
then
f
∈
Γ(W,
S)
if
and
only
if
(respectively,
implies
that)
f
can
be
written
étale
locally
(on
W
)
in
the
form
(az
+
b)/(cz
+
d),
where
a,
b,
c,
d
are
biformal
constants
and
d
is
invertible.
It
is
clear
that
if
S
⊆
O
U
bi
is
a
pre-Schwarz
structure
on
U
log
,
then
S
is
contained
in
a
unique
Schwarz
structure
S
a
⊆
O
U
bi
on
U
log
,
which
we
refer
to
as
the
Schwarz
structure
associated
to
S.
If
S
is
a
Schwarz
structure,
then
we
shall
denote
by
S
×
⊆
S
(respectively,
L
S
)
the
subsheaf
consisting
locally
of
functions
of
the
form
(az
+b)/(cz
+d),
where:
(1)
z
is
a
b
a
biformal
uniformizer
belonging
to
S;
(2)
d
is
invertible;
and
(3)
is
an
invertible
c
d
S
×
.
Thus
S
×
,
L
S
,
matrix
of
biformal
constants
(respectively,
b
=
0).
We
let
L
×
S
=
L
S
×
and
L
×
S
are
all
pre-Schwarz
structures.
We
shall
call
L
S
(respectively,
L
S
)
the
sheaf
of
biformal
uniformizers
(respectively,
pseudo-uniformizers)
of
S.
Let
G
→
U
be
the
group
scheme
PGL
2
,
and
let
B
⊆
G
be
the
subgroup
scheme
which
is
the
standard
Borel
subgroup
of
PGL
2
,
i.e.,
the
image
of
the
lower
triangular
matrices.
33
First
Properties
of
Schwarz
Structures
Proposition
1.3.
The
subsheaf
L
×
S
⊆
S
consisting
of
biformal
uniformizers
of
S
forms
a
B-torsor
B
S
→
U
.
Proof.
This
follows
immediately
from
the
definition
of
a
Schwarz
structure.
The
action
of
a
0
B
is
given
by
associating
to
a
biformal
uniformizer
z
and
a
matrix
(where
a,
c,
d
c
d
are
biformal
constants,
and
a,
d
are
invertible)
the
biformal
uniformizer
az/(cz
+
d).
Note
that
every
B-torsor
T
→
U
naturally
defines
a
P
1
-bundle
with
a
given
section
(by
taking
the
quotient
of
P
1
×
U
T
modulo
the
diagonal
action
of
B,
where
B
acts
on
P
1
by
means
of
affine
transformations
that
fix
zero;
the
section
is
the
image
of
the
zero
section
of
P
1
).
We
shall
refer
to
the
P
1
-bundle
P
S
→
U
associated
to
B
S
→
U
as
the
P
1
-bundle
associated
to
the
Schwarz
structure
S.
We
denote
by
σ
S
:
U
→
P
S
the
natural
section
(arising
from
the
fact
that
the
structure
group
is
B
rather
than
G).
Proposition
1.4.
Let
S
be
a
Schwarz
structure
on
U
log
.
Then
P
S
∼
=
P(J
/J
[3]
),
and
σ
S
∗
τ
P
S
/U
∼
=
(J
/J
[2]
)
∨
∼
=
τ
U
log
/S
log
.
In
particular,
if
U
→
S
is
proper,
then
the
height
of
σ
S
with
respect
to
τ
P
S
/U
is
−deg(ω
U
log
/S
log
).
Proof.
One
sees
by
construction
(e.g.,
by
writing
out
transition
functions)
that
the
sheaf
of
nonzero
relative
rational
functions
of
relative
degree
one
(as
in
[EGA
IV],
§20)
for
P
S
→
U
that
vanish
at
σ
is
naturally
isomorphic
to
L
×
S
.
Thus,
by
considering
Taylor
expansions
out
to
second
order
terms,
we
get
an
isomorphism
O
P
S
(−σ
S
)/O
P
S
(−3σ
S
)
∼
=
J
/J
[3]
(here
we
use
that
p
is
odd).
On
the
other
hand,
by
multiplying
and
then
taking
the
residue
at
σ,
we
obtain
a
natural
duality
between
O
P
S
(−σ
S
)/O
P
S
(−3σ
S
)
and
π
∗
ω
P
S
/U
(3σ
S
),
where
π
:
P
S
→
U
is
the
natural
projection.
Also,
note
that
via
this
duality,
the
filtration
induced
by
π
∗
ω
P
S
/U
(2σ
S
)
⊆
π
∗
ω
P
S
/U
(3σ
S
)
on
O
P
S
(−σ
S
)/O
P
S
(−3σ
S
)
is
the
filtration
defined
by
the
submodule
O
P
S
(−2σ
S
)/O
P
S
(−3σ
S
).
Since
P
S
is
clearly
naturally
isomorphic
to
the
projectivization
of
π
∗
ω
P
S
/U
(3σ
S
),
we
thus
obtain
the
result.
Crystalline
Schwarz
Structures
and
Monodromy
Let
S
be
a
Schwarz
structure
on
U
log
.
We
would
like
to
associate
to
S
a
subsheaf
(in
the
category
of
sets)
of
O
U
tr
,
which
we
shall
call
S
12
as
follows.
We
work
locally.
Thus,
we
assume
that
there
exists
a
biformal
uniformizer
z
∈
Γ(U,
S).
We
consider
the
triformal
function
z
12
defined
by
j
12
(z),
where
j
12
is
the
natural
map
O
U
bi
→
O
U
tr
given
by
inclusion
on
the
first
two
factors.
Then
we
let
S
12
be
the
sheaf
of
all
functions
which
étale
locally
can
be
written
in
the
form
(az
12
+
b)/(cz
12
+
d),
where
a,
b,
c,
d
are
triformal
constants
and
d
is
invertible.
Note
that
the
definition
of
S
12
does
not
depend
on
the
choice
of
z,
34
so
everything
glues
together,
and
we
obtain
the
subsheaf
S
12
of
O
U
tr
over
our
original
U
.
On
the
other
hand,
we
also
have
a
subsheaf
S
13
of
O
U
tr
defined
in
the
same
way
as
S
12
,
except
with
the
roles
of
2
and
3
reversed.
Definition
1.5.
We
shall
say
that
the
Schwarz
structure
S
is
a
crystalline
Schwarz
structure
on
U
log
if
the
two
subsheaves
S
12
and
S
13
of
O
U
tr
coincide.
Let
S
be
a
crystalline
Schwarz
structure
on
U
log
.
Then
we
shall
say
that
S
has
nilpotent
monodromy
if
for
every
marked
point
s
:
T
→
V
(with
V
→
U
étale),
there
exists
a
biformal
uniformizer
z
∈
S(V
)
and
a
section
a
∈
ω
U
log
/S
log
(V
)
such
that
the
image
of
(dz)
−
i
R
(a)
(where
“d”
is
the
exterior
derivative
on
the
right)
in
O
U
bi
⊗
O
U
s
∗
ω
U
log
/S
log
is
zero.
Remark.
Of
course,
one
may
also
phrase
the
definition
of
a
crystalline
Schwarz
structure
as
follows.
First,
note
that
O
U
bi
,
together
with
its
right-hand
sided
O
U
-algebra
structure
and
standard
logarithmic
connection,
forms
a
quasi-coherent
crystal
of
algebras
A
on
the
crystalline
site
of
U
log
/S
log
.
Then
a
crystalline
Schwarz
structure
is
a
subsheaf
of
the
sheaf
A
on
the
crystalline
site
of
U
log
/S
log
satisfying
certain
properties.
Since
this
point
of
view
is
only
formally
different
from
the
point
of
view
of
Definition
1.5,
we
shall
use
these
two
points
of
view
interchangeably
in
what
follows.
log
Let
us
suppose
that
S
is
a
Schwarz
structure
on
U
.
Let
H
⊆
G
be
the
open
a
b
subscheme
consisting
of
matrices
of
the
form
,
where
d
is
invertible.
Note
that
c
d
H
is
stable
under
the
action
by
B
from
the
right.
Thus,
we
can
take
the
quotient
of
H
×
U
B
S
(by
the
diagonal
action
of
B)
to
obtain
a
fiber
bundle
H
S
→
U
with
fibers
locally
isomorphic
to
H
→
U
.
Similarly,
we
also
obtain
a
G-torsor
G
S
→
U
.
It
now
follows
immediately
from
the
definitions
that
the
sheaf
defined
on
the
étale
site
of
U
by
H
S
is
naturally
isomorphic
to
S
×
.
Thus,
if
we
assume
that
the
Schwarz
structure
S
is
crystalline,
we
see
that
we
get
a
natural
isomorphism
between
the
two
pull-backs
of
H
S
→
U
via
i
L
,
i
R
:
O
U
→
O
U
bi
,
i.e.,
we
get
a
logarithmic
connection
∇
H
S
on
H
S
→
U
.
By
basic
facts
about
fiber
bundles,
this
gives
a
logarithmic
connection
∇
G
S
on
G
S
→
U
and
a
logarithmic
connection
∇
P
S
on
P
S
→
U
,
as
well.
Thus,
in
summary,
to
every
crystalline
Schwarz
structure
S,
we
have
associated
a
natural
P
1
-bundle
with
section
and
logarithmic
connection
(P
S
→
U
;
σ
S
:
U
→
P
S
;
∇
P
S
).
Moreover,
it
follows
from
the
definition
of
the
connection
∇
P
S
that
by
differentiating
σ
S
by
means
of
∇
P
S
,
we
get
an
isomorphism
τ
U
log
/S
log
∼
=
σ
S
∗
τ
P
S
/U
,
which
is
called
the
Kodaira-Spencer
morphism.
Indeed,
to
see
that
this
morphism
is,
indeed,
an
isomorphism,
it
suffices
to
realize
that
if,
locally
on
U
,
one
takes
a
biformal
uniformizer
z,
the
difference
j
12
(z)
−
j
13
(z)
generates
j
23
(
J
)O
U
tr
.
In
addition
to
the
fact
that
the
Kodaira-Spencer
morphism
is
an
isomorphism,
the
logarithmic
connection
∇
P
S
has
another
special
property:
If
S
has
nilpotent
monodromy,
then
we
can
make
more
explicit
the
way
in
which
this
monodromy
acts.
Indeed,
let
us
recall
from
[Kato],
§6,
that
if
s
:
S
→
U
is
any
marked
point,
then
there
exists
a
unique
subsheaf
M
s
of
(O
U
bi
⊗
O
U
,s
−1
O
S
)/O
S
which
is
isomorphic
to
O
S
and
annihilated
by
the
monodromy
operator
of
the
standard
logarithmic
connection
on
O
U
bi
.
(Locally,
35
this
subsheaf
is
generated
by
log(1
−
δ),
where
δ
=
1
−
(
1⊗t
t⊗1
),
and
t
is
a
local
generator
of
the
ideal
defining
s.)
This
subsheaf
M
s
thus
defines
a
section
q
s
:
S
→
P(J
/J
[3]
)
that
lies
over
s.
Then
it
follows
from
these
observations,
plus
Proposition
1.4,
that
Proposition
1.6.
If
S
is
a
crystalline
Schwarz
structure
on
U
log
with
nilpotent
mon-
odromy,
then
under
the
isomorphism
P
S
∼
=
P(J
/J
[3]
)
of
Proposition
1.4,
q
s
is
fixed
by
the
monodromy
action
on
P
S
∼
=
P(J
/J
[3]
)
at
s.
Correspondence
with
P
1
-bundles
So
far,
from
a
crystalline
Schwarz
structure,
we
have
constructed
a
P
1
-bundle
with
section
and
logarithmic
connection
(satisfying
certain
properties).
We
can
go
the
other
way,
as
well.
Suppose
we
are
given
a
P
1
-bundle
with
section
and
logarithmic
connection
(π
:
P
→
U
;
σ
:
U
→
P
;
∇
P
)
such
that
the
Kodaira-Spencer
morphism
obtained
by
differentiating
σ
via
∇
P
gives
an
isomorphism
τ
U
log
/S
log
∼
=
σ
∗
τ
P/U
.
Let
P
L
and
P
R
denote
the
pull-backs
of
π
:
P
→
U
via
i
L
,
i
R
:
O
U
→
O
U
bi
,
respectively.
Then
the
connection
∇
P
defines
an
O
U
bi
-linear
isomorphism
Ξ
:
P
L
∼
=
P
R
.
Thus,
we
have
a
commutative
diagram:
Ξ
P
L
−→
⏐
⏐
L
π
U
bi
id
−→
P
R
⏐
⏐
R
π
U
bi
Let
σ
L
(respectively,
σ
R
)
denote
the
result
of
base-changing
σ
via
i
L
(respectively,
i
R
).
Then
by
applying
(σ
L
)
∗
,
we
can
pull-back
functions
on
P
L
to
biformal
functions
on
U
log
.
Let
R
denote
the
étale
sheaf
of
degree
≤
1
relative
rational
functions
(as
in
[EGA
IV],
§20)
on
P
relative
to
π
:
P
→
U
(i.e.,
the
divisor
of
poles
is
flat
of
degree
≤
1
over
U
)
that
are
regular
in
a
neighborhood
of
the
image
of
σ.
Let
α
R
:
P
R
→
P
denote
the
natural
projection.
Then
it
is
easy
to
see
that
Ξ
−1
(α
R
)
−1
(R)
defines
a
sheaf
of
functions
on
P
L
that
are
regular
in
a
neighborhood
of
Im(σ
L
),
so
we
can
consider
the
subsheaf
S
of
biformal
functions
on
U
log
which
is
the
image
of
(σ
L
)
−1
Ξ
−1
(α
R
)
−1
(R).
One
checks
immediately
that
S
defines
a
Schwarz
structure,
and,
moreover,
that
since
the
connection
∇
P
is
necessarily
integrable
(since
the
dimension
of
U
over
S
is
one),
S
is
automatically
crystalline.
Thus,
in
summary,
we
have
the
crystalline
analogue
of
Theorem
2
of
[Gunning]:
Theorem
1.7.
If
f
:
U
→
S
is
as
above,
then
there
is
a
natural
one-to-one
correspondence
between
crystalline
Schwarz
structures
on
U
log
and
isomorphism
classes
of
P
1
-bundles
with
section
and
logarithmic
connection
(π
:
P
→
U
;
σ
:
U
→
P
;
∇
P
)
on
U
log
whose
associated
Kodaira-Spencer
morphism
is
an
isomorphism.
Moreover,
under
this
correspondence,
the
crystalline
Schwarz
structures
with
nilpotent
monodromy
correpond
precisely
to
the
triples
such
that
∇
P
has
nilpotent
monodromy
at
the
marked
points.
36
Proof.
We
have
already
defined
maps
going
in
either
direction.
Thus,
it
suffices
to
see
that
these
maps
are
inverse
to
each
other.
Now
it
is
easy
to
see
that
if
we
start
with
a
P
1
-bundle
with
section
and
logarithmic
connection
as
above,
construct
the
associated
crystalline
Schwarz
structure
S,
and
then
from
that
the
associated
P
1
-bundle
P
S
with
section
σ
S
and
logarithmic
connection
∇
P
S
,
then
we
get
back
our
original
data.
Thus,
it
suffices
to
show
that
the
map
that
associates
a
P
1
-bundle
with
section
and
connection
to
a
crystalline
Schwarz
structure
is
injective.
Let
S
and
S
be
crystalline
Schwarz
structures
on
U
log
.
Suppose
that
we
are
given
a
horizontal
isomorphism
α
between
P
S
and
P
S
that
takes
σ
S
to
σ
S
.
Then
α
induces
an
isomorphism
α
B
of
the
B-torsors
B
S
and
B
S
.
Let
A
be
the
O
U
-algebra
O
U
bi
via
∼
×
the
morphism
i
R
.
Since
as
sheaves
with
B-action,
B
S
∼
=
L
×
S
and
B
S
=
L
S
,
we
get
an
×
×
isomorphism
α
L
:
L
×
S
→
L
S
,
which,
by
mapping
a
biformal
uniformizer
z
∈
Γ(U,
L
S
)
(where
U
→
X
is
étale)
to
the
biformal
uniformizer
α
L
(z)
∈
Γ(U,
L
×
S
)
⊆
Γ(U,
A),
defines
an
automorphism
α
A
of
the
PD-O
U
-algebra
A
that
preserves
the
augmentation
μ
:
A
→
O
U
.
Moreover,
it
follows
from
the
horizontality
of
α
that
α
A
is
horizontal
with
respect
to
the
standard
logarithmic
connection
on
A.
On
the
other
hand,
it
is
immediate
that
A
does
admit
any
nontrivial
horizontal
automorphisms
(as
a
PD-O
U
-algebra)
that
preserve
μ.
Thus,
α
A
is
the
identity,
and
hence,
S
and
S
must
be
the
same
subsheaf
of
A.
This
completes
the
proof
of
the
first
statement.
The
last
statement
follows
directly
from
the
definitions.
Schwarz
Structures
and
Square
Differentials
We
would
like
to
use
Theorem
1.7
to
exhibit
the
space
of
Schwarz
structures
as
a
torsor
over
the
square
differentials.
Let
π
:
P
→
U
be
a
P
1
-bundle.
Then
we
shall
denote
by
Ad(P
)
the
vector
bundle
on
U
(of
rank
three
with
trivial
determinant)
given
by
π
∗
τ
P/U
.
When
we
consider
marked
points,
it
is
not
enough
just
to
deal
with
P
1
-bundles;
we
must
deal
with
P
1
-bundles
equipped
with
parabolic
structures,
as
in
[Sesh].
Thus,
if
our
divisor
D
of
marked
points
is
given
by
sections
p
1
,
.
.
.
,
p
r
:
S
→
U
,
we
make
the
following
Definition
1.8.
A
P
1
-bundle
with
parabolic
structure
on
U
log
is
defined
to
be
a
P
1
-bundle
π
:
P
→
U
,
together
with
sections
q
i
:
S
→
P
lying
over
p
i
.
A
rank
two
vector
bundle
with
parabolic
structure
on
U
log
is
a
rank
two
vector
bundle
E,
together
with
a
parabolic
structure
on
P(E).
Let
(π
:
P
→
U
;
q
1
,
.
.
.
,
q
r
)
be
a
P
1
-bundle
with
parabolic
structure
on
U
log
.
Then
we
define
the
subsheaf
Ad
q
(P
)
⊆
Ad(P
)
to
be
the
sheaf
of
sections
that
vanish
at
the
q
i
’s.
We
define
Ad
c
(P
)
⊆
Ad
q
(P
)
to
be
the
subsheaf
of
sections
that
vanish
to
second
order
(in
the
relative
coordinate
for
π)
at
the
q
i
’s.
Suppose
that
we
are
given
a
section
σ
:
U
→
P
that
avoids
all
the
q
i
.
Let
L
=
σ
∗
ω
P/U
.
Then
Ad(P
)
gets
a
filtration
0
=
F
2
(Ad(P
))
⊆
F
1
(Ad(P
))
⊆
F
0
(Ad(P
))
⊆
F
−1
(Ad(P
))
=
Ad(P
)
given
by
considering
sections
of
τ
P/U
that
vanish
to
first
or
second
order
at
σ.
Thus,
for
Ad(P
),
we
have:
37
F
1
∼
=
L;
F
0
/F
1
∼
=
O
U
;
F
−1
/F
0
∼
=
L
−1
This
filtration
induces
filtrations
on
Ad
q
(P
)
and
Ad
c
(P
).
The
subquotients
are
easily
seen
to
be
the
following:
For
Ad
q
(P
),
we
have:
F
1
∼
=
L(−D);
F
0
/F
1
∼
=
O
U
;
F
−1
/F
0
∼
=
L
−1
For
Ad
c
(P
),
we
have:
F
1
∼
=
L(−D);
F
0
/F
1
∼
=
O
U
(−D);
F
−1
/F
0
∼
=
L
−1
log
Often,
L
∼
=
ω
U/S
.
Thus,
for
computational
purposes,
it
is
convenient
to
note
that
log
ω
U/S
(−D)
is
none
other
than
the
relative
dualizing
sheaf
of
the
morphism
f
:
U
→
S.
Now
let
us
assume
that
π
:
P
→
U
is
given
by
P(J
/J
[3]
),
with
the
section
σ
given
by
J
/J
[3]
→
J
/J
[2]
,
and
the
q
i
given
by
the
sections
“q
s
”
defined
in
the
paragraph
preceding
Proposition
1.6.
Let
∇
P
be
a
logarithmic
connection
whose
Kodaira-Spencer
morphism
at
σ
is
the
identity
and
whose
monodromy
at
the
marked
points
is
nilpotent
and
fixes
the
q
i
.
(It
is
not
difficult
to
see
that
such
∇
P
always
exist
étale
locally
on
U
.)
Then
any
other
such
logarithmic
connection
∇
P
on
P
→
U
is
given
by
adding
to
∇
P
a
log
.
On
the
other
hand,
the
quadruples
(π;
σ;
q
i
;
∇
P
)
and
section
of
F
0
(Ad
c
(P
))
⊗
O
U
ω
U/S
(π;
σ;
q
i
;
∇
P
)
are
isomorphic
if
and
only
if
∇
P
can
be
obtained
from
∇
P
by
applying
an
automorphism
α
of
(π;
σ;
q
i
)
that
preserves
the
conormal
bundle
to
σ
(since
both
Kodaira-
Spencer
morphisms
are
the
identity).
Such
an
automorphism
α
is
given
by
a
section
of
F
1
(Ad
q
(P
)).
The
effect
of
such
an
automorphism
α
on
the
connection
∇
P
is
given
log
obtained
by
applying
the
morphism
by
adding
to
∇
P
the
section
of
Ad
c
(P
)
⊗
O
U
ω
U/S
log
Ad(∇
P
)
:
F
1
(Ad
q
(P
))
→
F
0
(Ad
c
(P
))
⊗
O
U
ω
U/S
(induced
by
the
connection
∇
P
)
to
α.
Thus,
we
obtain
that
the
set
of
isomorphism
classes
of
quadruples
(π;
σ;
q
i
;
∇
P
)
that,
relative
to
the
bijection
of
Theorem
1.7,
correspond
to
crystalline
Schwarz
structures
with
nilpotent
monodromy
are
a
torsor
over
the
cokernel
of
Ad(∇
P
).
On
the
other
hand,
by
looking
at
the
explicit
representations
of
the
subquotients
of
the
filtrations
on
Ad
q
(P
)
and
Ad
c
(P
)
(given
in
the
preceding
paragraph),
and
using
the
fact
that
the
Kodaira-Spencer
morphism
for
∇
P
at
σ
is
an
isomorphism,
we
obtain
that
log
Coker(Ad(∇
P
))
∼
=
(ω
U/S
)
⊗2
(−D)
(We
remark
that
here
one
uses
the
fact
that
p
is
odd,
for
when
one
computes
Ad(∇
P
)
from
∇
P
,
certain
factors
of
2
appear,
and
in
order
to
get
the
above
isomorphism,
one
needs
for
those
factors
of
2
to
be
invertible.)
In
other
words,
we
have
proven
the
following
result:
38
Theorem
1.9.
The
étale
sheaf
of
crystalline
Schwarz
structures
on
U
log
with
nilpotent
log
⊗2
)
(−D).
monodromy
is
naturally
a
torsor
over
the
sheaf
(ω
U/S
Normalized
P
1
-bundles
with
Connection
Let
us
consider
the
P
1
-bundle
π
:
P
=
P(J
/J
[3]
)
→
U
,
and
section
σ
:
U
→
P
given
by
J
/J
[3]
→
J
/J
[2]
,
without
any
connection.
Now,
just
as
in
the
proof
of
Propo-
sition
1.4,
by
taking
residues,
we
obtain
a
natural
duality
between
O
P
(−σ)/O
P
(−3σ)
and
π
∗
ω
P/U
(3σ)
that
respects
the
natural
filtrations
on
the
two
bundles.
Let
Q
=
P((J
/J
[3]
)
∨
).
As
U
-schemes,
we
may
identify
Q
and
P
.
Let
O
Q
(1)
denote
the
line
bundle
obtained
from
the
definition
of
the
projectivization;
thus
π
∗
O
Q
(1)
=
(J
/J
[3]
)
∨
.
Let
L
=
π
∗
{O
Q
(1)⊗
O
Q
τ
P/U
(−3σ)}.
Then
L
is
a
line
bundle
on
U
and
(J
/J
[3]
)
∨
=
L⊗
O
U
π
∗
ω
P/U
(3σ).
Thus,
we
obtain
a
natural
isomorphism
O
P
(−σ)/O
P
(−3σ)
∼
=
(J
/J
[3]
)⊗
O
U
L
that
respects
filtrations.
If
we
then
look
at
the
quotients
of
both
sides
by
their
respective
rank
one
subbundles
(that
make
up
the
filtrations),
we
obtain
an
isomorphism
between
log
∼
log
ω
U/S
=
(J
/J
[2]
)
⊗
L
∼
=
ω
U/S
⊗
L.
That
is,
we
get
a
natural
trivial-
=
O
P
(−σ)/O
P
(−2σ)
∼
ization
O
U
∼
=
L
of
L.
In
summary,
we
see
that
without
any
connection,
we
have
constructed
a
natural
filtration-preserving
isomorphism:
γ
:
O
P
(−σ)/O
P
(−3σ)
∼
=
J
/J
[3]
Now
let
us
suppose
that
we
have
a
logarithmic
connection
∇
P
on
π
whose
Kodaira-
Spencer
morphism
at
σ
is
an
isomorphism.
Then
we
get
a
commutative
diagram
like
the
one
preceding
Theorem
1.7.
Pulling
back
by
α
R
,
then
Ξ,
and
finally
by
σ
L
,
we
thus
see
that
∇
P
induces
an
isomorphism:
ζ(∇
P
)
:
O
P
(−σ)/O
P
(−3σ)
∼
=
J
/J
[3]
Now
we
saw
above
(Theorem
1.7)
that
∇
P
defines
a
Schwarz
structure.
But
one
“loose
end”
relative
to
the
statement
of
Theorem
1.7
is
that
although
Schwarz
structures
have
no
automorphisms,
triples
consisting
of
projective
bundles
with
a
section
and
a
connection
can
have
automorphisms.
These
automorphisms
were
the
cause
of
the
phenomenon
(observed
just
before
the
statement
of
Theorem
1.9)
that
many
different
∇
P
can
give
rise
to
the
same
Schwarz
structure.
Thus,
it
is
convenient
to
have
some
sort
of
notion
of
a
“normalized
∇
P
”
such
that
each
Schwarz
structure
arises
from
a
unique
normalized
∇
P
.
We
choose
the
normalization
as
follows:
Definition
1.10.
We
say
that
∇
P
is
normalized
if
γ
and
ζ(∇
P
)
are
inverse
to
each
other.
Now
as
a
formal
consequence
of
this
definition,
we
observe
that
we
obtain
the
following
normalized
version
of
Theorem
1.7:
39
Theorem
1.11.
If
f
:
U
→
S
is
as
above,
then
there
is
a
natural
one-to-one
correspon-
dence
between
crystalline
Schwarz
structures
on
U
log
and
normalized
logarithmic
connec-
tions
on
the
P
1
-bundle
π
:
P(J
/J
[3]
)
→
U
whose
associated
Kodaira-Spencer
morphism
at
the
section
σ
:
U
→
P
(defined
by
J
/J
[3]
→
J
/J
[2]
)
is
an
isomorphism.
Moreover,
under
this
correspondence,
the
crystalline
Schwarz
structures
with
nilpotent
monodromy
correpond
precisely
to
the
triples
such
that
∇
P
has
nilpotent
monodromy
at
the
marked
points.
The
Schwarzian
Derivative
Before
proceeding,
it
is
interesting
to
note
that,
as
the
name
suggests,
a
crystalline
Schwarz
structure
S
allows
one
to
define
a
Schwarzian
derivative
d
S
,
as
follows.
Let
×
ω
U
log
/S
log
⊆
ω
U
log
/S
log
denote
the
subsheaf
consisting
of
sections
that
locally
generate
#
⊆
O
U
be
the
subsheaf
consisting
of
functions
φ
such
ω
U
log
/S
log
as
an
O
U
-module.
Let
O
U
×
that
dφ
is
a
section
of
ω
U
log
/S
log
⊆
ω
U
log
/S
log
.
Then
our
Schwarzian
derivative
will
be
a
morphism
of
sheaves
of
sets:
#
⊗2
d
S
:
O
U
→
ω
U
log
/S
log
#
Let
θ
be
a
section
of
O
U
over
some
étale
V
→
U
.
Let
us
denote
by
j
θ
∈
J
/J
[3]
(V
)
the
2-
jet
of
θ
(i.e.,
the
Taylor
expansion
out
to
second
order,
modulo
the
constant
term).
By
the
#
definition
of
O
U
,
the
image
of
j
θ
in
J
/J
[2]
(V
)
is
a
local
generator
of
the
sheaf
J
/J
[2]
.
By
Proposition
1.6,
j
θ
then
defines
a
section
s
θ
:
V
→
P
S
.
Taking
the
Kodaira-Spencer
map
of
this
section
then
defines
an
O
V
-linear
morphism
from
τ
U
log
/S
log
to
the
conormal
bundle
to
s
θ
,
which
is
simply
ω
U
log
/S
log
.
This
O
V
-linear
morphism
is
thus
given
by
multiplication
⊗2
by
a
section
of
ω
U
log
/S
log
,
which
we
take
to
be
d
S
(θ).
A
simple
calculation
reveals
that
Proposition
1.12.
If
(as
in
Theorem
1.9)
one
modifies
the
Schwarz
structure
S
by
log
⊗2
)
(−D)](U
)
to
obtain
a
Schwarz
structure
S
,
adding
the
square
differential
δ
∈
[(ω
U/S
then
d
S
(θ)
=
d
S
(θ)
+
δ.
We
also
have
a
biformal
version
of
the
Schwarzian
derivative.
Namely,
we
let
O
#
bi
be
U
the
subsheaf
of
O
U
bi
consisting
of
biformal
functions
φ
that
are
of
the
form
u
+
c,
where
u
is
a
biformal
uniformizer,
and
c
is
a
biformal
constant.
Then
we
get
a
morphism
of
sheaves
of
sets:
#
⊗2
⊗2
L
d
bi
S
:
O
bi
→
(ω
U
log
/S
log
)
=
ω
U
log
/S
log
⊗
O
U
O
U
bi
def
U
defined
as
follows:
If
θ
is
a
section
of
O
#
bi
over
some
étale
V
→
U
,
we
let
j
θ
be
the
U
section
of
J
/J
[3]
⊗
O
U
O
U
bi
which
is
the
2-jet
of
θ.
Thus,
j
θ
defines
a
section
of
s
θ
of
P
S
L
40
⊗2
L
whose
Kodaira-Spencer
map
is
given
by
multiplication
by
a
section
of
(ω
U
log
/S
log
)
,
which
bi
we
take
as
d
bi
S
(θ).
Note
that
d
S
(i
L
(θ))
=
i
L
(d
S
(θ)),
and
that
if
we
modify
the
Schwarz
structure
by
adding
a
square
differential
δ,
then
bi
d
bi
S
(θ)
=
d
S
(θ)
+
i
L
(δ)
Remark.
In
the
Appendix
to
this
Chapter,
we
show
that
the
definition
just
given
for
the
Schwarzian
derivative
coincides
with
one-half
the
classical
Schwarzian
derivative,
when
U
is
the
projective
line.
For
the
biformal
version
of
the
Schwarzian,
we
have
an
analogue
of
the
classical
result
that
the
Schwarzian
vanishes
exactly
on
the
formal
functions
that
make
up
the
projective
structure
of
a
Riemann
surface.
Indeed,
let
Ξ
:
P
S
L
→
P
S
R
be
the
isomorphism
defined
by
the
connection
∇
P
S
;
let
α
R
:
P
S
R
→
P
S
be
the
natural
projection;
and
let
def
ζ
:
U
bi
=
Spec(O
U
bi
)
→
P
S
be
the
morphism
obtained
by
composing
σ
S
L
:
U
bi
→
P
S
L
with
Ξ
and
then
α
R
.
Since
the
definition
of
d
S
is
functorial,
d
S
applied
to
a
function
pulled
back
by
ζ
is
ζ
−1
of
the
“d
S
”
computed
for
projective
bundles
in
the
Appendix,
i.e.,
one-half
the
classical
Schwarzian.
Thus,
if
θ
is
a
section
of
S,
then
θ
is
the
pull-back
by
ζ
of
a
(degree
≤
1)
relative
rational
function
for
P
S
→
U
,
so
d
bi
S
(θ)
=
0.
bi
Conversely,
suppose
that
d
bi
S
(θ)
=
0.
Then
the
statement
that
d
S
(θ)
=
0
means
L
that
s
θ
is
a
horizontal
section
of
P
S
.
Thus
it
follows
from
the
definition
of
a
connection,
together
with
the
Poincaré
Lemma
in
crystalline
cohomology
(see,
e.g.,
[Kato],
§6
for
the
log
version)
that
Ξ(s
θ
)
is
the
pull-back
via
α
R
of
a
section
t
θ
of
P
S
.
Now
(after
possible
étale
localization),
we
can
find
a
(degree
≤
1)
relative
rational
function
φ
for
P
S
→
U
whose
2-jet
at
σ
S
is
given
by
the
section
t
θ
.
Since
ζ
maps
the
diagonal
in
U
bi
to
σ
S
R
,
and
def
the
formation
of
2-jets
is
functorial,
it
thus
follows
that
the
2-jet
of
ψ
=
ζ
−1
(φ)
defines
a
section
s
ψ
of
P
S
L
which
is
equal
to
s
θ
when
restricted
to
the
diagonal
U
⊆
U
bi
.
But
since
both
s
ψ
and
s
θ
are
horizontal,
they
must
be
equal.
The
biformal
functions
ψ
and
θ
thus
have
2-jets
that
define
the
same
“line”
in
J
/J
[3]
.
Let
z
be
a
local
coordinate
on
U
.
Let
us
denote
by
successive
primes
the
derivatives
of
biformal
functions
(i.e.,
taken
on
the
left)
with
respect
to
z.
Then
we
obtain
that
ψ
and
θ
are
both
invertible
biformal
functions
such
that
ψ
·
θ
=
ψ
·
θ
.
It
thus
follows
that
(ψ
/θ
)
=
0,
so
θ
=
a
ψ
,
where
a
is
an
invertible
biformal
constant.
Thus,
θ
=
a
ψ
+
b,
where
b
is
a
biformal
constant.
Since,
by
construction
ψ
∈
S(V
),
it
follows
that
θ
∈
S(V
).
Thus,
we
obtain
the
following
“crystalline
Schwarzian
Poincaré
Lemma:”
Theorem
1.13.
If
θ
is
a
biformal
function,
then
d
bi
S
(θ)
=
0
if
and
only
if
θ
is
a
section
of
S.
41
§2.
Indigenous
Bundles
In
this
Section,
we
globalize
the
local
considerations
of
§1,
and
are
thus
led
to
introduce
“indigenous
bundles”
(as
in
[Gunning]).
Let
S
log
be
a
fine
log
scheme,
whose
underlying
scheme
is
connected
noetherian.
Let
f
log
:
X
log
→
S
log
be
proper,
geometrically
connected,
and
locally
stable
of
dimension
one.
(Note
that
the
first
two
conditions
are
actually
conditions
on
the
underlying
scheme
morphism
f
.)
We
assume
that
the
fibers
of
f
:
X
→
S
have
arithmetic
genus
g
≥
0,
and
exactly
r
≥
0
marked
points
(as
in
Definition
1.1
–
note
that
these
may
only
be
defined
étale
locally,
however).
Basic
Definitions
and
Examples
If
2g
−
2
+
r
≥
1,
then
let
M
g,r
be
the
moduli
stack
of
stable
curves
of
genus
g,
with
r
marked
points,
over
Z,
and
let
ζ
:
C
→
M
g,r
be
the
universal
curve,
with
its
r
marked
points
s
1
,
.
.
.
,
s
r
:
M
g,r
→
C.
Note
that
M
g,r
has
a
natural
log
structure
given
log
by
the
divisor
at
infinity.
Denote
the
resulting
log
stack
M
g,r
.
Also,
by
taking
the
divisor
which
is
union
of
the
s
i
and
the
pull-back
of
the
divisor
at
infinity
of
M
g,r
,
we
get
a
log
structure
on
C;
we
call
the
resulting
log
stack
C
log
.
Also,
ζ
:
C
→
M
g,r
extends
naturally
log
to
a
morphism
of
log
stacks
ζ
log
:
C
log
→
M
g,r
.
Definition
2.1.
We
shall
say
that
f
log
:
X
log
→
S
log
is
stable
if
there
exists
a
classifying
log
morphism
φ
log
:
S
log
→
M
g,r
such
that
X
log
∼
=
S
log
×
M
log
C
log
.
g,r
Ultimately,
we
shall
be
concerned
mainly
with
the
case
where
f
log
is
stable,
but
it
is
useful
to
realize
that
the
definition,
as
well
as
many
of
the
first
properties,
of
indigenous
bundles
can
be
made
without
these
assumptions.
Let
π
:
P
→
X
be
a
P
1
-bundle.
If
σ
:
X
→
P
is
a
section,
then
we
call
the
canonical
height
of
σ
the
number
12
deg
X/S
(σ
∗
τ
P/X
),
where
deg
X/S
denotes
the
relative
degree
over
S
of
a
line
bundle
on
X,
and
τ
P/X
is
the
relative
tangent
bundle
of
π.
If
∇
P
is
a
logarithmic
connection
on
P
,
then
we
call
the
morphism
τ
X
log
/S
log
→
σ
∗
τ
P/X
given
by
differentiating
σ
by
means
of
∇
P
the
Kodaira-Spencer
morphism
at
σ
relative
to
∇
P
.
Often,
instead
of
dealing
with
P
1
-bundles
with
logarithmic
connections,
it
will
be
more
convenient
to
deal
vector
bundles:
Thus,
let
E
be
a
vector
bundle
equipped
with
a
logarithmic
connection
∇
E
,
whose
rank
is
two
and
whose
determinant
is
trivial.
Then
Theorem
1.7
motivates
the
following
Definition
2.2.
We
shall
say
that
(π
:
P
→
X,
∇
P
)
is
an
indigenous
bundle
on
X
log
if
the
monodromy
at
the
marked
points
(which
exist
étale
locally)
is
nilpotent,
and
there
exists
a
section
σ
:
X
→
P
of
π
such
that
the
Kodaira-Spencer
morphism
at
σ
with
respect
to
∇
P
is
an
isomorphism.
We
shall
say
that
(E,
∇
E
)
is
an
indigenous
vector
bundle
42
on
X
log
if
the
associated
P
1
-bundle
with
logarithmic
connection
(P(E)
→
X,
∇
P(E)
)
is
an
indigenous
P
1
-bundle.
We
shall
say
that
P
→
X
(respectively,
E)
is
intrinsic
if
there
exists
a
logarithmic
connection
∇
P
(respectively,
∇
E
)
on
P
→
X
(respectively,
E)
that
makes
(P
→
X,
∇
P
)
(respectively,
(E,
∇
E
))
indigenous.
We
shall
say
that
P
→
X
(respectively,
E)
is
locally
intrinsic
if
it
is
intrinsic
étale
locally
on
S.
Thus,
in
the
vector
bundle
case,
(E,
∇
E
)
is
indigenous
if
∇
E
has
nilpotent
monodromy
at
the
marked
points,
and
there
exists
a
rank
one
subbundle
F
0
(E)
⊆
E
such
that
the
Kodaira-Spencer
morphism
F
0
(E)
→
ω
X
log
/S
log
⊗
O
X
(E/F
0
(E))
(induced
by
∇
E
)
is
an
isomorphism.
So
far
we
have
been
discussing
the
hyperbolic
case
(2g
−
2
+
r
≥
1);
however,
one
can
make
the
same
definition
for
curves
that
are
not
hyperbolic.
Example
1.
Suppose
that
f
:
X
→
S
is
smooth,
with
no
marked
points,
and
that
g
=
0.
Thus,
f
is
a
P
1
-bundle.
Then
the
P
1
-bundle
given
by
X
×
S
X
→
X
has
a
natural
trivial
connection,
together
with
a
natural
section,
the
diagonal
section.
It
is
trivial
to
see
that
this
triple
satisfies
the
required
properties
for
an
indigenous
bundle.
Example
2.
Suppose
that
f
log
:
X
log
→
S
log
has
no
marked
points,
and
that
its
fibers
all
have
arithmetic
genus
one.
Then
consider
the
bundle
E
=
ω
X/S
⊕
O
X
(where
ω
X/S
is
the
relative
dualizing
sheaf).
Let
L
=
f
∗
ω
X/S
.
Thus,
L
is
a
line
bundle
on
S,
and
f
∗
L
∼
=
ω
X/S
.
In
particular,
there
exists
on
ω
X/S
a
“trivial
connection”
∇
ω
obtained
from
tensoring
the
trivial
connection
on
O
X
with
f
∗
L.
Let
∇
E
be
the
connection
on
E
which
is
the
direct
sum
of
∇
ω
and
the
trivial
connection
on
O
X
.
Let
∇
E
be
the
connection
on
E
given
by
adding
to
∇
E
the
section
of
End(E)
⊗
ω
X/S
given
by
projecting
E
→
ω
X/S
∼
=
(0,
O
X
)
⊗
ω
X/S
⊆
E
⊗
ω
X/S
.
Then
one
checks
easily
that
if
we
take
(P,
∇
P
)
=
P(E,
∇
E
),
and
σ
:
X
→
P
to
be
given
by
(ω
X/S
,
0)
⊆
E,
then
we
obtain
an
indigenous
bundle
on
X
log
.
Example
3.
Let
S
log
=
Spec(Z)
(with
the
trivial
log
structure);
X
log
=
M
1,1
(the
moduli
stack
of
one-pointed
curves
of
genus
one
over
Z),
with
its
natural
log
structure.
Let
E
be
the
vector
bundle
of
rank
two
on
X
which
is
the
first
de
Rham
cohomology
module
of
the
universal
one-pointed
curve
of
genus
one.
Then
E
has
a
natural
logarithmic
connection
∇
E
,
the
so-called
“Gauss-Manin
connection.”
There
is
also
a
natural
Hodge
def
filtration
F
1
(E)
⊆
E,
which
defines
a
section
σ
:
X
→
P
=
P(E).
The
pair
(E,
∇
E
)
forms
the
prototypical
example
of
an
indigenous
bundle
on
X
log
.
First
Properties
We
now
proceed
to
examine
basic
properties
of
such
bundles.
43
Proposition
2.3.
If
f
log
:
X
log
→
S
log
and
h
log
:
Y
log
→
S
log
are
as
stipulated
at
the
beginning
of
the
Section,
and
ζ
log
:
Y
log
→
X
log
is
a
log
étale
morphism
of
log
schemes
over
S
log
that
sends
marked
points
to
marked
points,
then
the
pull-back
via
ζ
log
of
any
indigenous
bundle
is
again
indigenous.
Proof.
This
follows
from
the
definitions.
Proposition
2.4.
If
π
:
P
→
X
is
intrinsic,
then
the
section
σ
:
X
→
P
is
of
canonical
height
1
−
g
−
12
r.
If
X
log
→
S
log
is
a
stable
curve
(so,
in
particular,
2g
−
2
+
r
≥
1),
then
σ
is
the
unique
section
of
π
of
canonical
height
1
−
g
−
12
r.
We
shall
refer
to
σ
as
the
Hodge
section
of
π
:
P
→
X.
Proof.
The
fact
that
the
canonical
height
of
σ
is
1
−
g
−
12
r
follows
from
the
fact
that
the
Kodaira-Spencer
morphism
is
an
isomorphism.
Now
suppose
that
X
log
→
S
log
is
stable.
Let
us
first
assume
that
S
is
the
spectrum
of
an
algebraically
closed
field.
Suppose
that
σ
:
X
→
P
also
has
canonical
height
1−g
−
12
r.
Then
it
follows
that
its
restriction
to
some
irreducible
component
of
X
has
negative
canonical
height.
Since
the
restriction
of
σ
∗
τ
P/X
to
any
irreducible
component
has
negative
degree,
it
follows
immediately
from
considering
intersection
numbers
on
P
,
together
with
the
definition
of
“canonical
height,”
that
there
cannot
exist
two
distinct
sections
of
negative
canonical
height
over
that
irreducible
com-
ponent.
Thus,
σ
and
σ
must
agree
over
that
irreducible
component.
Now
if
there
are
any
other
irreducible
components
in
X,
then
σ
must
have
negative
canonical
height
over
some
other
irreducible
component
of
X,
in
order
for
its
canonical
height
over
all
of
X
to
be
1
−
g
−
12
r.
Thus,
repeating
this
argument
shows
that
σ
=
σ
.
Finally,
let
us
observe
that
the
space
of
deformations
of
σ
is
given
by
H
0
(X,
σ
∗
τ
P/X
),
which
is
zero,
since
σ
∗
τ
P/X
has
negative
degree
on
every
irreducible
component
of
X.
The
result
for
general
S
then
follows
immediately
from
this
by
deformation
theory.
Now
let
us
assume
for
the
rest
of
the
Section
(unless
stated
otherwise)
that
there
exists
an
odd
prime
p
which
is
nilpotent
on
S,
together
with
a
PD-ideal
I
⊆
O
S
.
Proposition
2.5.
If
π
:
P
→
X
is
intrinsic,
then
P
∼
=
P(J
/J
[3]
)
(where
J
defines
the
diagonal
in
X
bi
).
Moreover,
for
any
connection
on
π
that
makes
it
indigenous,
the
mon-
odromy
at
a
marked
point
s
:
S
→
X
fixes
the
section
q
s
:
S
→
P(J
/J
[3]
)
of
Proposition
1.6.
Proof.
This
follows
from
Theorem
1.7
and
Propositions
1.3
and
1.6.
Note
that
the
second
statement
uses
the
fact
that
p
is
odd.
Proposition
2.6.
Suppose
that
the
number
of
marked
point
plus
nodes
on
any
geometric
irreducible
component
of
a
fiber
of
X
→
S
is
even.
Let
(π
:
P
→
X,
∇
P
)
be
indigenous
on
44
X
log
.
Then
étale
locally
on
S,
there
exists
an
indigenous
vector
bundle
(E,
∇
E
)
on
X
log
whose
projectivization
is
(π
:
P
→
X,
∇
P
).
Moreover,
such
an
(E,
∇
E
)
is
unique
up
to
tensor
product
with
a
line
bundle
with
connection
(L,
∇
L
)
on
X
whose
square
is
trivial.
Proof.
Consider
the
relative
anticanonical
bundle
τ
P/X
on
P
.
By
Proposition
2.5,
(after
étale
localization
on
S)
there
exists
a
line
bundle
G
on
P
whose
square
is
τ
P/X
.
Now
let
us
note
that
since
the
construction
of
the
anticanonical
bundle
is
canonical,
it
follows
that
the
connection
∇
P
on
the
P
1
-bundle
induces
a
connection
on
the
polarized
P
1
-bundle
(π
:
P
→
X,
τ
P/X
).
Moreover,
since
the
“moduli
space”
of
line
bundles
G
whose
square
is
τ
P/X
is
étale
over
X,
it
follows
that
the
connection
∇
P
on
the
P
1
-bundle
π
:
P
→
X
in
fact
induces
a
connection
on
the
polarized
P
1
-bundle
(π
:
P
→
X,
G).
Thus,
we
def
get
a
connection
∇
E
on
E
=
π
∗
G.
Moreover,
on
P
,
we
have
a
natural
exact
sequence
0
→
ω
P/X
→
(π
∗
E)
⊗
G
−1
→
O
P
→
0,
which
induces
an
isomorphism
det(π
∗
E)
∼
=
O
P
,
hence
an
isomorphism
det(E)
∼
=
O
X
,
which
is
easily
seen
to
be
horizontal.
Finally,
it
is
clear
that
the
projectivization
of
(E,
∇
E
)
is
isomorphic
to
(π
:
P
→
X,
∇
P
).
Now
suppose
that
both
(E,
∇
E
)
and
(E
,
∇
E
)
have
the
same
projectivization
(π
:
P
→
X,
∇
P
).
Then
E
defines
a
line
bundle
G
on
P
whose
square
is
τ
P/X
and
such
that
∇
P
induces
a
connection
on
the
polarized
P
1
-bundle
(π
:
P
→
X,
G).
Similarly,
E
defines
a
line
bundle
G
on
P
.
Since
we
have
horizontal
isomorphisms
G
⊗2
∼
=
τ
P/X
and
(G
)
⊗2
∼
=
τ
P/X
,
−1
it
follows
that
if
we
let
L
=
π
∗
((G
)
⊗
G),
then
L
gets
a
natural
connection
∇
L
such
that
the
square
of
(L,
∇
L
)
is
trivial.
Moreover,
(E,
∇
E
)
∼
=
(E
,
∇
E
)
⊗
(L,
∇
L
).
This
completes
the
proof.
In
summary,
the
above
Proposition
tells
us
that
(under
the
evenness
assumption)
up
to
étale
localization
on
the
base,
it
is
essentially
the
same
thing
to
give
an
indigenous
P
1
-
bundle
or
an
indigenous
vector
bundle.
Thus,
in
the
future,
we
shall
frequently
simply
speak
of
“indigenous
bundles.”
The
same
goes
for
intrinsic
bundles.
Existence
and
de
Rham
Cohomology
The
next
step
is
to
prove
the
existence
of
indigenous
bundles,
and
to
parametrize
them.
We
begin
with
the
proof
of
existence.
For
the
rest
of
this
Section,
we
shall
assume
that
f
log
is
stable.
Thus,
in
particular,
2g
−
2
+
r
≥
1.
Theorem
2.7.
For
any
r-pointed
stable
curve
X
log
→
S
log
of
genus
g,
the
P
1
-bundle
P(J
/J
[3]
)
is
locally
intrinsic.
Proof.
From
Theorem
1.9,
we
know
that
the
obstruction
to
the
existence
of
a
crystalline
Schwarz
structure
with
nilpotent
monodromy
on
X
log
(locally
on
S)
is
given
by
a
section
log
⊗2
log
⊗2
of
R
1
f
∗
(ω
X/S
)
(−D)
over
S.
On
the
other
hand,
by
Serre
duality,
R
1
f
∗
(ω
X/S
)
(−D)
45
is
isomorphic
to
the
dual
of
f
∗
τ
X
log
/S
log
=
0,
since
the
curve
is
stable.
The
Theorem
now
follows
form
Proposition
1.6
and
Theorem
1.7.
Next
we
wish
to
compute
the
de
Rham
cohomology
of
the
P
1
-bundle
with
parabolic
structure
(π;
q
i
).
Note
that
the
exterior
differential
operator
maps
Ad(P
)
(respectively,
Ad
q
(P
))
into
Ad
q
(P
)
(respectively,
Ad
c
(P
)).
We
define
the
parabolic
de
Rham
cohomology
(respectively,
with
compact
supports)
of
Ad(P
)
to
be
the
hypercohomology
of
the
complex
log
log
(respectively,
Ad
q
(P
)
→
Ad
c
(P
)
⊗
ω
X/S
).
Ad(P
)
→
Ad
q
(P
)
⊗
ω
X/S
Theorem
2.8.
Let
(P,
∇
P
)
be
an
indigenous
bundle
on
an
r-pointed
stable
curve
f
log
:
X
log
→
S
log
of
genus
g.
Then
the
de
Rham
cohomology
of
Ad(P
)
with
its
natural
connec-
tion
(induced
by
∇
P
)
is
as
follows:
(1)
For
cohomology
without
compact
supports,
we
have
(f
DR
)
∗
(Ad(P
))
=
R
2
(f
DR
)
∗
(Ad(P
))
=
0;
and
we
have
a
natural
exact
sequence
log
⊗2
)
(−D)
→
R
1
(f
DR
)
∗
(Ad(P
))
→
R
1
f
∗
τ
X
log
/S
log
→
0
0
→
f
∗
(ω
X/S
(2)
For
cohomology
with
compact
supports,
we
have
(for
all
i
≥
0)
a
natural
isomorphism
R
i
(f
DR
)
c,∗
(Ad(P
))
∼
=
R
i
(f
DR
)
∗
(Ad(P
))
In
particular,
(P
;
∇
P
)
has
no
nontrivial
automorphisms.
Proof.
To
compute
the
de
Rham
cohomology,
one
uses
the
long
exact
cohomology
se-
quences
induced
by
the
filtrations
considered
above,
plus
the
fact
that
the
Kodaira-Spencer
morphism
is
an
isomorphism.
Now
let
α
be
an
automorphism.
Since
(f
DR
)
∗
(Ad(P
))
=
0,
all
infinitesimal
automorphisms
must
vanish,
so
we
may
work
over
an
algebraically
closed
field.
By
passing
to
a
tamely
ramified
covering
of
X
ramified
only
at
the
marked
points
and
nodes,
we
may
assume
that
the
hypotheses
of
Proposition
2.6
are
satisfied.
Then
let
L
be
a
line
bundle
on
P
whose
square
is
τ
P/X
.
Since
L
∼
=
O
P
(σ)
⊗
O
S
M
(for
a
line
bundle
M
on
S),
and
α
always
preserves
σ,
it
follows
that
α
preserves
L.
Thus,
α
arises
from
a
horizontal
section
of
End(π
∗
L)
=
O
X
⊕
Ad(P
),
hence
is
induced
by
multiplying
π
∗
L
by
a
section
of
O
S
.
Thus,
α
is
the
identity,
as
desired.
Finally,
combining
what
we
have
done
in
this
Section
with
Theorem
1.7,
we
obtain:
Corollary
2.9.
Let
f
log
:
X
log
→
S
log
be
an
r-pointed
stable
curve
of
genus
g.
Then
the
set
of
crystalline
Schwarz
structures
on
X
with
nilpotent
monodromy
is
in
one-to-one
46
correspondence
the
set
of
isomorphism
classes
of
indigenous
P
1
-bundles
on
X
log
.
More-
over,
the
functor
that
assigns
to
T
log
→
S
log
the
set
of
crystalline
Schwarz
structures
with
log
⊗2
)
(−D).
nilpotent
monodromy
on
X
T
log
=
X
log
×
S
log
T
log
is
a
torsor
over
f
∗
(ω
X/S
Indigenous
Bundles
of
Restrictable
Type
Let
f
i
log
:
X
i
log
→
S
log
(for
i
=
1,
.
.
.
,
n)
be
an
r
i
-pointed
smooth
curve
of
genus
g
i
(where
2g
i
−
2
+
r
i
≥
1
for
all
i).
Suppose
that
we
are
given
a
graph
Γ
consisting
of
n
vertices,
numbered
1
through
n.
Let
E
i
be
the
set
of
edges
of
the
i
th
vertex.
Suppose
further
that
we
are
given
an
injection
λ
i
:
E
i
→
{1,
.
.
.
,
r
i
}.
Then
we
can
glue
together
the
curves
f
i
log
:
X
i
log
→
S
log
to
form
an
r-pointed
stable
curve
f
log
:
X
log
→
S
log
of
genus
g
in
such
a
way
that
the
dual
graph
of
f
log
is
given
by
Γ,
that
is:
(1)
vertex
i
corresponds
to
f
i
log
:
X
i
log
→
S
log
,
an
irreducible
component
of
X
log
;
(2)
if
is
an
edge
running
from
vertex
i
to
vertex
j
such
that
λ
i
(
)
=
a
and
λ
j
(
)
=
b,
then
corresponds
to
a
node
on
X
log
obtained
by
gluing
together
X
i
log
at
the
a
th
marked
point
to
X
j
log
at
the
b
th
marked
point;
(3)
g
and
r
can
be
computed
combinatorially
from
Γ,
the
g
i
’s,
the
r
i
’s
and
the
λ
i
’s.
:
X
i
log
→
X
log
be
the
inclusion
of
X
i
log
into
X
log
as
one
of
the
irreducible
Let
μ
log
i
components.
Now
let
us
suppose
that
we
are
given
an
indigenous
bundle
(π
:
P
→
X;
∇
P
)
on
X
log
.
log
∗
Then
it
is
not
necessarily
the
case
that
(μ
log
i
)
(π
:
P
→
X;
∇
P
)
will
be
indigenous
on
X
i
.
The
problem
is
that
since
in
general,
marked
points
of
X
i
log
might
be
sent
to
nodes
of
X
log
(and
not
to
marked
points),
there
is
no
reason
why
the
monodromy
at
such
marked
points
of
X
i
log
should
be
nilpotent.
We
therefore
make
the
following
log
∗
Definition
2.10.
If
the
(μ
log
for
all
i,
then
we
i
)
(π
:
P
→
X;
∇
P
)
are
indigenous
on
X
i
say
that
(π
:
P
→
X;
∇
P
)
is
of
restrictable
type.
Now
let
us
suppose
that
we
are
given
indigenous
bundles
Π
i
=
(π
i
:
P
i
→
X
i
;
∇
P
i
)
on
X
i
log
.
Note
that
for
each
marked
point
s
:
S
→
X
i
of
an
X
i
log
,
s
∗
P
i
has
a
canonical
trivialization
as
a
P
1
-bundle
given
by
considering:
(1)
the
Hodge
section
σ
i
:
X
i
→
P
i
(pulled-back
by
s);
(2)
the
trivialization
of
s
∗
(σ
i
∗
ω
P
i
/X
i
)
given
by
the
residue
map;
and
(3)
the
section
q
s
:
S
→
s
∗
P
i
of
Proposition
1.6.
47
It
thus
follows
that
we
can
glue
together
the
Π
i
’s
by
means
of
this
canonical
trivialization
at
the
marked
points
to
obtain
an
indigenous
bundle
Π
=
(π
:
P
→
X;
∇
P
)
on
X
log
.
Moreover,
by
construction,
Π
is
of
restrictable
type.
Also,
we
can
clearly
reverse
the
procedure:
Namely,
if
we
start
with
an
indigenous
bundle
Π
on
X
log
of
restrictable
type,
we
can
reconstruct
Π
by
restricting
to
the
X
i
log
’s,
and
then
regluing,
in
the
fashion
described
in
the
preceding
paragraph.
Now
let
us
define
n
(f
i
)
∗
ω
⊗2
log
def
E
=
X
i
i=1
/S
log
(−D
i
)
where
D
i
is
the
divisor
of
marked
points
on
X
i
.
Then
we
have
the
following
result:
Proposition
2.11.
The
étale
sheaf
on
S
of
isomorphism
classes
of
indigenous
P
1
-bundles
of
restrictable
type
on
X
log
is
a
torsor
over
the
vector
bundle
E.
Note
that
the
rank
of
E
is
given
by
than
3g
−
3
+
r.
n
i=1
(3g
i
−
3
+
r
i
),
which,
in
general,
is
strictly
less
§3.
The
Obstruction
to
Global
Intrinsicity
In
§2,
we
saw
that
the
P
1
-bundle
P(J
/J
[3]
)
is
locally
intrinsic
on
M
g,r
.
In
this
Section,
we
study
the
obstruction
(which,
in
general,
is
nonzero)
to
it
being
globally
intrinsic
over
all
of
M
g,r
.
The
main
point
is
a
computation
in
Hodge
cohomology
which,
in
many
respects,
is
similar
to
that
of
[Falt3],
Lemma
IV.4.
Since,
however,
it
is
not
literally
the
same
as
[Falt3],
and
certain
technical
aspects
of
the
computation
are
different,
we
provide
a
complete
proof
here.
Introduction
of
Cohomology
Classes
We
shall
work
over
a
field
K
of
characteristic
zero,
say
Q
p
,
until
we
state
otherwise.
Since
we
are
only
interested
in
certain
intersection
numbers,
the
base
field
is
essentially
irrelevant.
Let
us
consider
the
universal
r-pointed
stable
curve
of
genus
g,
ζ
:
C
→
M
g,r
(over
K).
We
would
like
to
consider
various
cohomology
classes
on
C
and
M
g,r
.
The
cohomology
theory
that
we
will
use
is
Hodge
theory,
so
all
cohomology
classes
are
to
be
understood
as
being
Hodge-theoretic.
Let
π
:
P
=
P(J
/J
[3]
)
→
C
be
the
P
1
-bundle
which,
as
we
saw
in
§2,
is
locally
intrinsic.
Let
F
=
Ad(P
).
Thus,
F
has
a
filtration
whose
subquotients
are
given
by:
F
1
∼
=
L;
F
0
/F
1
∼
=
O
C
;
F
−1
/F
0
∼
=
L
−1
48
where
L
=
ω
C
log
/M
log
.
Let
η
=
c
1
(L),
the
first
Chern
class
of
L.
Then
we
have:
c
1
(F)
=
0.
g,r
On
the
other
hand,
the
second
Chern
class
of
F
is
given
by:
c
2
(F)
=
−η
2
2
Let
r
us
compute
ζ
∗
η
.
Let
D
i
⊆
C
(where
i
=
1,
.
.
.
,
r)
be
the
marked
points.
Let
D
=
i=1
D
i
.
We
shall
write
[D
i
];
[D]
for
the
respective
cohomology
classes
on
C.
Let
ξ
=
c
1
(ω
C/M
g,r
).
Thus,
η
=
ξ
+
[D].
Since
different
D
i
’s
do
not
intersect,
we
have
[D
i
]
·
[D
j
]
=
0
if
i
=
j.
Also,
by
“taking
the
residue,”
we
see
that
ζ
∗
{(ξ
+
[D
i
])
·
[D
i
]}
=
0,
r
for
all
i.
Thus,
ζ
∗
{(ξ
+
[D])
·
[D]}
=
0.
Let
ψ
i
=
ζ
∗
(ξ
·
[D
i
]);
ψ
=
i=1
ψ
i
;
θ
=
ζ
∗
ξ
2
.
Then
we
obtain:
ζ
∗
η
2
=
ζ
∗
{(ξ
+
[D])
·
ξ}
=
θ
+
ψ
Now
one
knows
from
[AC]
that
for
g
≥
3,
the
restrictions
of
the
classes
θ
and
ψ
to
M
g,r
are
linearly
independent.
We
summarize
this
in
a
Lemma:
Lemma
3.1.
We
have,
on
M
g,r
,
ζ
∗
η
2
=
θ
+
ψ.
In
particular,
if
g
≥
3,
then
(ζ
∗
η
2
)|
M
g,r
is
nonzero.
We
shall
see
below
that
ζ
∗
c
2
(F)
can
be
related
to
the
obstruction
to
the
existence
of
a
global
indigenous
bundle
on
C.
Thus,
once
we
have
done
this,
we
will
have
proven
that
this
obstruction
is
given
by
the
relatively
computable
cohomology
class
−ζ
∗
η
2
on
M
g,r
.
Computation
of
the
Second
Chern
Class
Let
us
first
observe
that
P
|
D
has
a
canonical
trivialization
as
a
P
1
-bundle
given
by
using:
(1)
the
Hodge
section
σ
:
C
→
P
(restricted
to
D);
(2)
the
trivialization
of
(σ
∗
ω
P/C
)|
D
given
by
the
residue
map;
and
(3)
the
section
q
s
:
D
→
P
×
C
D
of
Proposition
1.6.
Let
us
denote
by
C
c
log
the
log
stack
obtained
by
letting
C
be
the
underlying
stack
and
log
taking
for
the
log
structure
the
pull-back
of
the
log
structure
of
M
g,r
via
ζ.
Thus,
we
have
an
exact
sequence
on
C:
0
→
ζ
∗
Ω
M
log
→
Ω
C
c
log
→
ω
C/M
g,r
→
0
g,r
49
where
the
first
two
sheaves
of
differentials
are
over
K.
In
the
future,
we
shall
think
of
this
exact
sequence
as
defining
a
one-step
filtration
on
Ω
C
c
log
.
For
i,
j
≥
0,
let
us
define
for
any
O
C
-module
G:
def
H
c
i,j
(C,
G)
=
H
i
(C,
G
⊗
O
C
∧
j
Ω
C
c
log
)
This
cohomology
is
a
sort
of
cohomology
with
compact
supports
outside
D.
Thus,
by
using
the
canonical
trivialization
of
P
|
D
referred
to
above,
we
see
that
the
global
obstruction
to
the
existence
of
a
logarithmic
connection
on
π
:
P
→
C
(for
the
log
structure
of
C
log
)
which
has
normalized
nilpotent
monodromy
at
D
defines
a
class
κ
∈
H
c
1,1
(C,
Ad(P
)).
Now,
by
taking
the
trace
of
the
square
of
κ,
we
get
a
class
in
tr(κ
2
)
∈
H
c
2,2
(C,
O
C
).
If
we
then
apply
log
def
ζ
∗
,
we
get
a
class
ζ
∗
tr(κ
2
)
∈
H
1,1
(M
g,r
)
=
H
1
(M
g,r
,
Ω
M
log
).
On
the
other
hand,
let
us
g,r
denote
by:
S
g,r
→
M
g,r
the
Ω
M
log
-torsor
defined
by
looking
at
the
crystalline
Schwarz
structures
with
nilpotent
g,r
log
monodromy
on
C
(as
in
Corollary
2.9).
This
torsor
thus
defines
a
class
Σ
∈
H
1,1
(M
g,r
).
The
goal
of
this
subsection
is
to
prove
the
following:
log
Lemma
3.2.
We
have
the
following
equality
of
classes
in
H
1,1
(M
g,r
):
Σ
=
12
ζ
∗
tr(κ
2
).
Now
let
us
note
that
H
c
1,1
(C,
Ad(P
))
has
two
one-step
filtrations:
one
arising
from
the
Leray-Serre
spectral
sequence
applied
to
ζ,
and
the
other
arising
from
the
filtration
defined
above
on
Ω
C
c
log
.
Thinking
in
these
terms,
we
see
that
we
get
a
morphism:
φ
00
:
H
c
1,1
(C,
Ad(P
))
→
H
0
(M
g,r
,
R
1
ζ
∗
(Ad(P
)
⊗
ω
C/M
g,r
))
Now
since
we
know
that
P
→
C
admits
a
connection
of
the
desired
type
on
the
fibers
of
ζ,
it
follows
that
φ
00
(κ)
=
0.
Next
let
us
consider
the
natural
morphism:
φ
10
:
H
c
1,1
(C,
Ad(P
))
→
H
1
(C,
Ad(P
)
⊗
ω
C/M
g,r
)
Since
φ
00
(κ)
=
0,
it
follows
from
considering
the
Leray-Serre
spectral
sequence
that
φ
10
(κ)
lies
in
H
1
(M
g,r
,
ζ
∗
(Ad(P
)
⊗
ω
C/M
g,r
))
→
H
1
(C,
Ad(P
)
⊗
ω
C/M
g,r
).
In
fact,
by
consid-
ering
only
normalized
connections
(as
in
Definition
1.10),
we
can
say
more.
Namely,
it
follows
that
φ
10
(κ)
is
actually
the
image
under
the
morphism
H
1
(M
g,r
,
ζ
∗
(ω
C
log
/M
log
⊗
g,r
ω
C/M
g,r
))
→
H
(M
g,r
,
ζ
∗
(Ad(P
)
⊗
ω
C/M
g,r
))
(induced
by
ω
C
log
/M
log
→
Ad(P
))
of
the
1
g,r
50
class
Σ
(regarded
as
a
class
in
H
1
(M
g,r
,
ζ
∗
(ω
C
log
/M
log
⊗
ω
C/M
g,r
))
by
means
of
the
tauto-
g,r
logical
isomorphism
Ω
log
∼
)).
log
⊗
ω
=
ζ
∗
(ω
M
g,r
C/M
g,r
C
log
/M
g,r
Now
let
us
consider
the
natural
morphism:
φ
01
:
H
c
1,1
(C,
Ad(P
))
→
H
0
(M
g,r
,
R
1
ζ
∗
(Ad(P
)
⊗
Ω
C
c
log
))
Since
φ
00
(κ)
=
0,
it
follows
that
the
section
φ
01
(κ)
of
R
1
ζ
∗
(Ad(P
)
⊗
Ω
C
c
log
)
lies
in
the
image
of
the
natural
map
ι
:
R
1
ζ
∗
(Ad(P
)
⊗
Ω
M
log
)
→
R
1
ζ
∗
(Ad(P
)
⊗
Ω
C
c
log
)
g,r
In
fact,
we
can
say
more.
Since
we
are
dealing
with
sheaves
on
M
g,r
,
we
can
compute
locally
on
M
g,r
.
Let
U
→
M
g,r
be
étale.
Let
∇
P
be
a
logarithmic
connection
with
normalized
nilpotent
monodromy
(relative
to
ζ
U
:
C
U
→
U
)
on
P
U
=
P
×
M
g,r
U
.
Then
the
obstruction
to
lifting
∇
P
to
a
logarithmic
connection
relative
to
C
U
→
Spec(K)
is
giving
by
subtracting
the
two
pull
backs
of
(P
U
→
U
;
∇
P
)
to
the
first
infinitesimal
neighborhood
Δ
U
of
the
diagonal
of
U
×
K
U
.
Note
that
it
only
makes
sense
to
compare
these
two
pull-backs
because
we
have
chosen
a
connection
∇
P
,
so
that
we
can
deal
with
crystals
on
Crys(C
U
/Δ
U
)
(where
the
structure
morphism
C
U
→
Δ
U
is
given
by
composing
ζ
U
:
C
U
→
U
with
the
diagonal
embedding
U
→
Δ
U
).
Thus,
the
difference
between
the
pull-backs
defines
a
section
δ
(over
U
)
of
Ω
M
log
|
U
⊗R
1
(ζ
U
)
DR,∗
(Ad(P
),
∇
P
).
Now
if
we
compose
the
projection
g,r
R
1
(ζ
U
)
DR,∗
(Ad(P
),
∇
P
)
→
R
1
(ζ
U
)
∗
τ
C
log
/U
log
with
δ,
we
get
a
morphism
U
(Ω
M
log
)
∨
|
U
=
Θ
M
log
|
U
→
R
1
(ζ
U
)
∗
τ
C
log
/U
log
g,r
U
g,r
Now
it
is
a
tautology
that
this
morphism
is
none
other
than
the
isomorphism
β
derived
log
from
deformation
theory
of
the
tangent
space
to
M
g,r
with
the
first
cohomology
group
of
the
relative
tangent
bundle
of
ζ
log
.
Thus,
in
summary,
we
have
proven
the
following
statement:
(*)
locally
on
M
g,r
,
φ
10
(κ)
is
the
image
under
ι
of
some
local
section
ν
of
R
1
ζ
∗
(Ad(P
)
⊗
Ω
M
log
)
whose
image
in
R
1
ζ
∗
(τ
C
log
/M
g,r
)
⊗
Ω
M
log
is
g,r
g,r
the
tautological
isomorphism
β.
We
are
now
ready
to
consider
ζ
∗
tr(κ
2
).
We
begin
by
using
the
observations
of
the
preceding
two
paragraphs
to
compute
what
happens
when
we
multiply
various
subquotients
of
the
two
filtrations
on
H
c
1,1
(C,
Ad(P
))
by
each
other:
(1)
If
we
multiply
two
elements
in
the
image
of
H
1
(M
g,r
,
ζ
∗
(Ad(P
))
⊗
Ω
M
log
),
and
take
the
trace,
we
get
a
(2,
2)-Hodge
cohomology
class
on
g,r
51
C
which
is
the
pull-back
of
such
a
class
on
M
g,r
;
thus,
if
we
apply
ζ
∗
to
such
a
product,
we
get
zero.
(2)
If
we
multiply
an
element
in
the
image
of
H
1
(M
g,r
,
ζ
∗
(Ad(P
))⊗Ω
M
log
)
g,r
by
an
element
in
the
image
of
H
0
(M
g,r
,
R
1
ζ
∗
(Ad(P
))⊗Ω
M
log
),
and
take
g,r
the
trace,
we
get
a
class
in
H
1
(M
g,r
,
R
1
ζ
∗
O
C
⊗
∧
2
Ω
M
log
);
since
there
g,r
are
no
factors
of
ω
C/M
g,r
in
the
wedge
product,
applying
ζ
∗
again
gives
zero.
(3)
If
we
multiply
two
elements
of
H
0
(M
g,r
,
R
1
ζ
∗
(Ad(P
))
⊗
Ω
M
log
),
we
g,r
get
zero
since
ζ
has
relative
dimension
one.
(4)
If
we
multiply
an
element
in
the
image
of
H
1
(M
g,r
,
ζ
∗
(Ad(P
))⊗Ω
M
log
)
g,r
by
φ
10
(κ)
∈
H
(M
g,r
,
ζ
∗
(Ad(P
)
⊗
ω
C/M
g,r
)),
and
take
the
trace,
we
get
zero,
since
we
are
taking
the
trace
of
the
product
of
a
nilpotent
section
of
Ad(P
)
with
a
section
of
the
same
Borel
subalgebra
of
Ad(P
).
1
(5)
If
we
square
φ
10
(κ)
∈
H
1
(M
g,r
,
ζ
∗
(Ad(P
)⊗ω
C/M
g,r
)),
we
get
zero
since
we
are,
in
effect,
squaring
nilpotent
sections
of
Ad(P
).
(6)
If
we
multiply
φ
10
(κ)
∈
H
1
(M
g,r
,
ζ
∗
(Ad(P
)
⊗
ω
C/M
g,r
))
by
φ
01
(κ)
∈
H
0
(M
g,r
,
R
1
ζ
∗
(Ad(P
)
⊗
Ω
C
c
log
)),
then
we
are,
in
effect,
multiplying
the
class
Σ
by
the
tautological
isomorphism
β,
so
that
we
obtain
Σ,
regarded
as
a
class
in
H
1
(M
g,r
,
Ω
M
log
).
g,r
Thus,
in
summary,
all
of
the
possible
contributions
are
zero,
except
for
the
last,
which
is
2Σ.
This
completes
the
proof
of
Lemma
3.2.
On
the
other
hand,
by
basic
linear
algebra,
c
2
(Ad(P
))
is
−2
tr(κ
2
),
so
we
see
that
we
have,
in
fact,
proven
the
following:
Lemma
3.3.
We
have
ζ
∗
c
2
(Ad(P
))
=
−4Σ.
Finally,
putting
this
together
with
Lemma
3.1,
we
see
that
we
have
explicitly
computed
the
class
Σ
in
terms
of
well-known
first
Chern
classes
of
line
bundles:
Theorem
3.4.
The
torsor
of
Schwarz
structures
defines
a
class
Σ
∈
H
1
(M
g,r
,
Ω
M
log
)
g,r
which
is
equal
to
14
(θ
+
ψ),
where
θ
=
ζ
∗
ξ
2
;
ψ
=
ζ
∗
(ξ
·
[D]);
ξ
=
c
1
(ω
C/M
g,r
);
and
[D]
is
the
cohomology
class
of
the
divisor
of
marked
points.
In
particular,
if
g
≥
3,
then
(in
characteristic
zero)
S
g,r
→
M
g,r
does
not
admit
any
sections,
i.e.,
there
are
no
canonical
Schwarz
structures
on
r-pointed
smooth
curves
of
genus
g
≥
3.
Remark.
Ideally,
it
would
be
nice
to
have
an
equality
of
classes
not
in
H
1
(M
g,r
,
Ω
M
log
),
g,r
but
in
some
sort
of
cohomology
with
compact
supports
“H
c
1
(M
g,r
,
Ω
M
log
)”
(which
should
g,r
52
be
isomorphic
to
H
1
(M
g,r
,
Ω
M
g,r
)).
In
order
to
do
this,
one
would
have
to
define
some
sort
of
appropriate
sense
in
which
Σ
is
compactly
supported,
i.e.,
one
would
have
to
define
some
sort
of
trivializations
of
S
g,r
→
M
g,r
at
infinity.
In
fact,
S
g,r
→
M
g,r
does
not
(in
general)
have
a
canonical
section
over
the
divisor
at
infinity.
However,
by
considering
indigenous
bundles
of
restrictable
type,
one
can
show
that,
so
to
speak,
“the
more
singular
a
curve
gets,
the
more
of
a
canonical
trivialization
one
has
for
S
g,r
→
M
g,r
.”
For
instance,
if
the
curve
is
totally
degenerate,
i.e.,
it
can
be
constructed
by
gluing
together
(as
at
the
end
of
§2)
a
number
of
copies
of
P
1
with
three
marked
points,
then
S
g,r
→
M
g,r
does
have
a
canonical
trivialization,
as
follows
immediately
from
Proposition
2.11
(since
then
the
indigenous
bundles
of
restrictable
type
form
a
torsor
over
the
zero
sheaf).
Thus,
in
some
sort
of
combinatorially
complicated
sense,
by
considering
indigenous
bundles
of
restrictable
type,
one
can
exhibit
Σ
as
a
cohomology
class
with
compact
supports.
Unfortunately,
however,
the
combinatorics
involved
get
rather
complicated
in
general,
so
we
shall
not
carry
this
out
explicitly,
except
in
the
case
when
the
dimension
of
M
g,r
is
one,
where
things
are
not
so
difficult.
The
Case
of
Dimension
One
In
this
case,
either
g
=
r
=
1
or
g
=
0,
r
=
4.
Let
D
∞
⊆
M
g,r
be
the
divisor
at
infinity.
Since
D
∞
is
zero-dimensional,
in
this
case
we
do
have
a
canonical
trivialization
t
∞
of
S
g,r
→
M
g,r
over
D
∞
.
Now
we
shall
give
a
new
definition
of
cohomology
with
compact
supports
that
takes
into
account
this
trivialization
t
∞
.
Let
ω
C
log
(respectively,
ω
(M
log
)
)
be
g,r
!
!
the
subsheaf
ω
C/M
g,r
(−D
∞
)
(respectively,
ω
M
log
(−D
∞
))
of
ω
C/M
g,r
(respectively,
ω
M
g,r
).
g,r
Let
Ω
C
log
be
the
inverse
image
of
ω
C
log
⊆
ω
C/M
g,r
via
the
morphism
Ω
C
c
log
→
ω
C/M
g,r
.
!
!
Thus,
we
have
an
exact
sequence:
0
→
ζ
∗
Ω
M
log
→
Ω
C
log
→
ω
C
log
→
0
g,r
!
!
which
defines
a
filtration
on
Ω
C
log
.
Then
for
i,
j
≥
0,
let
us
define
for
any
O
C
-module
G:
!
def
H
!
i,j
(C,
G)
=
H
i
(C,
G
⊗
O
C
∧
j
Ω
C
log
)
!
and
for
any
O
M
g,r
-module
H:
def
H
!
i,j
(M
g,r
,
H)
=
H
i
(M
g,r
,
H
⊗
O
M
g,r
∧
j
ω
(M
log
)
)
g,r
!
Note
that
the
push
forward
map
ζ
∗
naturally
defines
a
morphism
ζ
∗
:
H
!
2,2
(C,
O
C
)
→
H
!
1,1
(M
g,r
,
O
M
g,r
).
Now
the
obstruction
to
putting
a
logarithmic
connection
on
P
→
C
with
normalized
nilpotent
monodromy
at
the
marked
points
and
which
is
of
restrictable
53
type
at
infinity
is
given
by
a
class
κ
!
∈
H
!
1,1
(C,
Ad(P
)).
On
the
other
hand,
the
torsor
S
g,r
→
M
g,r
together
with
the
trivialization
t
∞
defines
a
class
Σ
!
∈
H
!
1,1
(M
g,r
,
O
M
g,r
).
By
the
same
proof
as
before,
we
have
Σ
!
=
12
ζ
∗
tr(κ
2!
).
(In
the
present
case,
however,
one
might
remark
that
in
the
six
types
of
product
considered
previously,
the
first
two
types
(numbered
(1)
and
(2))
of
product
vanish
all
the
more
trivially
since
they
involve
∧
2
ω
M
log
,
which
is
zero.)
Also,
just
as
before,
we
have
c
2
(Ad(P
))
=
−2
tr(κ
2!
).
Thus,
we
obtain:
g,r
Theorem
3.5.
In
H
!
1,1
(M
g,r
,
O
M
g,r
)
=
K,
we
have
Σ
!
=
1
ζ
∗
η
2
4
where
η
=
c
1
(ω
C
log
/M
log
).
g,r
Let
us
compute
ζ
∗
η
2
in
the
case
g
=
1,
r
=
1.
First
we
introduce
the
classes
θ
=
ζ
∗
ξ
2
;
ψ
=
ζ
∗
(ξ
·
[D]);
and
λ
=
c
1
(ζ
∗
ω
C/M
g,r
).
By
Grothendieck-Riemann-Roch,
θ
=
0.
On
the
other
hand,
sorting
through
the
definitions,
one
sees
that
ψ
=
λ.
Thus,
we
obtain
that
for
g
=
1,
r
=
1:
Σ
!
=
log
1
λ
4
log
log
Next
let
N
log
:
M
1,1
[2]
→
M
1,1
be
the
finite,
log
étale
covering
such
that
M
1,1
[2]
is
the
moduli
stack
that
parametrizes
elliptic
curves
with
level
structure
on
the
2-torsion
points.
Let
log
log
Λ
log
:
M
1,1
[2]
→
M
0,4
be
the
log
étale
morphism
given
by
sending
an
elliptic
curve
with
a
trivialization
of
its
two
torsion
to
the
four-pointed
curve
of
genus
zero
of
which
the
elliptic
curve
is
a
double
covering
(with
ramification
exactly
at
the
four
marked
points).
Moreover,
Λ
log
admits
a
log
section
over
any
double
covering
of
M
0,4
(since
the
obstruction
to
such
a
section
lies
in
2
(M
0,4
,
Z/2Z)).
Note
that
both
N
log
and
Λ
log
are,
in
fact,
defined
over
Z[
12
].
Also,
H
et
note
that
(over
Z[
12
]),
we
have
an
isomorphism
Λ
∗
S
0,4
∼
=
N
∗
S
1,1
obtained
by
pulling-back
and
pushing
forward
indigenous
bundles.
We
thus
obtain
the
following
result:
Theorem
3.6.
When
M
g,r
is
one-dimensional,
the
torsor
S
g,r
→
M
g,r
does
not
admit
any
section
which
is
equal
to
t
∞
over
D
∞
modulo
any
prime
≥
5.
54
Proof.
We
have
(Σ
!
)
1,1
=
14
λ
1,1
,
and
λ
1,1
=
12
c
1
(ω
M
log
),
so
(Σ
!
)
1,1
=
18
c
1
(ω
M
log
).
Thus,
1,1
∗
log
1,1
∗
is
log
étale,
and
N
(Σ
!
)
1,1
is
an
invertible
multiple
of
Λ
(Σ
!
)
0,4
,
it
suffices
to
since
N
show
that:
(1)
in
Hodge
cohomology
modulo
p
(for
a
prime
p
≥
5),
Λ
∗
c
1
(ω
M
log
)
=
0,4
N
∗
c
1
(ω
M
log
)
is
nonzero;
in
fact,
since
Λ
log
admits
a
section
over
any
1,1
log
double
covering
of
M
0,4
,
it
suffices
merely
to
show
that
c
1
(ω
M
log
)
is
0,4
nonzero
modulo
p;
(2)
the
formation
of
H
!
1,1
(M
0,4
,
O
M
0,4
)
commutes
with
base
change
modulo
p
(for
p
an
odd
prime).
But
both
(1)
and
(2)
follow
immediately
from
the
fact
that
M
0,4
is
just
P
1
,
with
D
∞
=
{0,
1,
∞}.
(Note
that
there
is
a
slight
subtlety
here
in
that
(2)
is
not
immediately
obvious
for
M
1,1
since
it
is
a
stack;
this
is
why
we
choose
to
verify
the
assertions
of
the
Theorem
by
means
of
M
0,4
.)
Appendix:
Relation
to
the
Complex
Analytic
Case
In
this
Appendix,
we
make
the
connection
between
the
theory
of
Schwarz
structures
discussed
here
and
the
classical
notion
of
the
Schwarzian
derivative
in
complex
analysis.
Let
K
be
an
algebraically
closed
field
of
characteristic
zero
(such
as,
for
instance,
the
complex
numbers
C).
Let
X
→
Spec(K)
be
a
smooth,
proper,
connected
curve.
Let
P
=
P(J
/J
[3]
)
be
the
usual
P
1
-bundle
on
X,
and
let
∇
P
be
a
connection
on
P
→
X
that
makes
it
indigenous.
Then
just
as
in
§1,
we
can
form
the
Schwarzian
derivative:
#
2
D
:
O
X
→
ω
X/K
Then
the
purpose
of
this
Appendix
is
to
show
that
when
X
=
P
1
;
z
is
the
standard
rational
2
is
trivialized
by
(dz)
2
,
then
D(φ)
is
given
(up
to
a
factor
of
function
on
P
1
;
and
ω
X/K
two)
by
the
classical
formula
for
the
Schwarzian
derivative.
First,
let
us
note
that
when
X
=
P
1
,
there
exists
only
one
connection
∇
P
on
π
:
P
→
X
that
makes
it
indigenous.
Indeed,
this
follows
from
the
fact
that
Ad(P
)
⊗
O
X
ω
X/K
has
no
sections
for
degree
reasons,
plus
the
fact
that
the
extension
0
→
J
[2]
/J
[3]
→
J
/J
[3]
→
J
/J
[2]
→
0
does
not
split
(since
the
extension
class
is
the
Hodge-theoretic
first
Chern
class
c
1
(ω
X/K
),
which
is
nonzero).
It
thus
follows
that
(P
;
∇
P
)
is
necessarily
isomorphic
to
the
indigenous
55
bundle
constructed
in
Example
1
of
§2.
Thus,
P
∼
=
X
×
K
X
(where
we
regard
the
projection
to
the
second
factor
as
the
structure
morphism
to
X);
let
us
fix
such
an
isomorphism
for
the
rest
of
the
discussion.
Also,
the
Kodaira-Spencer
morphism
at
the
diagonal
section
σ
Δ
:
X
→
X
×
K
X
∼
=
P
is
an
isomorphism.
Let
z
be
the
standard
rational
function
on
X.
Let
U
⊆
X
be
the
complement
of
infinity.
Thus,
z
is
regular
on
U
.
We
shall
work
mainly
on
U
.
For
i
=
1,
2,
let
p
i
:
P
∼
=
X
×
K
X
→
X
be
the
projection
to
the
i
th
factor.
Let
ζ
be
the
relative
rational
function
on
P
|
U
→
U
given
by
p
∗
1
(z)
−
p
∗
2
(z).
Let
us
d
.
Then
clearly,
simply
denote
by
∇
the
result
of
applying
∇
P
in
the
tangent
direction
dz
2
∇(ζ)
=
−1.
Let
η
=
1/ζ.
Thus,
∇(η)
=
η
.
Now
if
we
regard
P
as
P(J
/J
[3]
),
and
s
is
a
section
given
by
[a
dz,
b
(dz)
2
]
(where
a,
b
∈
O
X
(V
),
for
some
open
V
⊆
U
),
then
we
have
ζ(s)
=
a/b.
Indeed,
both
sides
of
this
equation
define
relative
rational
functions
on
P
|
U
→
U
.
The
right-hand
side
has
a
simple
pole
at
the
section
[dz,
0
·
(dz)
2
],
which,
by
computing
residues
as
in
the
definition
of
the
isomorphism
of
Proposition
1.4,
corresponds
to
the
section
∞
×
U
.
Thus,
both
sides
of
the
equation
have
a
simple
pole
at
∞
×
U
,
and
the
same
1-jet
at
σ
Δ
,
hence
are
equal.
In
particular,
η(s)
=
b/a.
#
(V
).
The
2-jet
j
φ
of
φ
is
given
by
φ
dz
+
12
φ
(dz)
2
.
Now
suppose
we
are
given
φ
∈
O
X
φ
Thus,
if
s
φ
is
the
section
of
P
over
V
that
is
defined
by
j
φ
,
we
have
η(s
φ
)
=
2φ
.
Thus,
we
compute:
(
φ
)
=
∇(η(s
φ
))
2φ
=
{∇(η)}(s
φ
)
+
η(∇(s
φ
))
=
η
2
(s
θ
)
+
D(φ)
(dz)
−2
=
(φ
)
2
+
D(φ)
(dz)
−2
2
4(φ
)
Expanding
the
derivative
on
the
left
and
rearranging
terms,
we
get:
φ
(φ
)
2
(φ
)
2
−
−
}
(dz)
2
2φ
2(φ
)
2
4(φ
)
2
1
φ
3(φ
)
2
=
{
−
}
(dz)
2
2
φ
2(φ
)
2
D(φ)
=
{
We
have
thus
shown
the
following
result:
Theorem
A.
On
P
1
,
the
Schwarzian
derivative
defined
at
the
end
of
§1
is
equal
to
one-half
the
classical
Schwarzian
derivative.
For
a
treatment
of
the
classical
Schwarzian
derivative,
we
refer
to
[Lehto].
56
Chapter
II:
Indigenous
Bundles
in
Characteristic
p
§0.
Introduction
In
this
Chapter,
we
study
indigenous
bundles
in
characteristic
p.
In
particular,
we
will
be
concerned
with
how
these
bundles
interact
with
Frobenius.
Our
main
tool
for
studying
this
interaction
will
be
the
p-curvature.
We
begin
in
§1
by
studying
FL-bundles,
which
are
a
special
kind
of
rank
two
vector
bundle
with
connection
on
a
curve
that
corresponds
to
a
lifting
of
the
curve
modulo
p
2
.
In
§2,
we
define
the
Verschiebung
map
on
indigenous
bundles
to
be
the
determinant
of
the
p-curvature
of
the
indigenous
bundle.
It
turns
out
that
(essentially)
indigenous
bundles
arise
from
FL-bundles
precisely
when
their
Verschiebung
vanishes.
Since
it
is
precisely
this
sort
of
indigenous
bundle
–
which
(following
[Katz])
we
call
nilpotent
–
that
corresponds
to
an
MF
∇
-object
in
the
sense
of
[Falt],
it
is
worthwhile
defining
and
studying
the
moduli
space
N
g,r
of
such
bundles.
In
order
to
study
N
g,r
,
we
make
two
fundamental
calculations
(Theorems
2.3
and
2.13)
concerning
the
Verschiebung.
The
first
tells
us
that
the
Verschiebung
is
finite
and
flat,
of
degree
p
3g−3+r
,
and
the
second
calculates
the
derivative
the
Verschiebung
in
terms
of
invariants
of
the
indigenous
bundle
which
are
easier
to
compute.
In
§3,
we
define
the
hyperbolic
(higher
genus)
analogue
of
an
ordinary
elliptic
curve:
namely,
we
say
that
a
hyperbolic
curve
is
hyperbolically
ordinary
if
it
admits
a
nilpotent
indigenous
bundle
at
which
the
derivative
of
the
Verschiebung
is
an
isomorphism.
Using
the
general
machinery
developed
in
§2,
we
then
do
a
number
of
computations
involving
totally
degenerate
curves
and
elliptic
curves
which
reveal
that:
(1)
the
hyperbolically
ordinary
locus
of
M
g,r
is
open
and
dense
(Corollary
3.8);
(2)
if
one
applies
the
definition
of
ordinariness
in
terms
of
indigenous
bun-
dles
to
the
case
of
elliptic
curves,
one
recovers
the
classical
definition
of
an
ordinary
elliptic
curve
(Theorem
3.11);
and
(3)
(at
least
if
g
≥
3,
and
p
is
sufficiently
large
then)
each
irreducible
component
of
N
g,r
that
passes
through
a
certain
canonical
nilpotent
indigenous
bundle
on
a
totally
degenerate
curve
has
degree
≥
2
over
M
g,r
;
thus,
there
is
no
canonical
choice
of
a
nilpotent
indigenous
bundle
on
a
generic
r-pointed
stable
curve
of
genus
g
(Proposition
3.13).
We
end
the
Chapter
with
the
observation
that
(3)
is
interesting
in
the
sense
that
it
con-
stitutes
a
deviation
from
the
behavior
that
one
might
expect
by
analogy
to
the
complex
case.
57
§1.
FL-Bundles
In
this
Section,
we
develop
the
theory
of
a
certain
kind
of
rank
two
bundle,
which
we
call
an
FL-bundle,
which
arises
from
looking
at
the
Cartier
isomorphism
of
a
curve.
It
turns
out
that
the
space
of
such
bundles
can
also
be
used
to
parametrize
the
infinitesimal
deformations
of
a
curve
to
Z/p
2
Z.
The
material
we
present
here
is
essentially
“well-known”
(see,
e.g.,
[Kato],
§4),
although
our
point
of
view
is
a
little
different.
Let
p
be
a
prime.
Let
S
be
a
noetherian
scheme
over
F
p
.
Let
us
assume
that
we
are
given
a
fine
log
structure
on
S,
and
let
us
denote
the
resulting
log
scheme
by
S
log
.
Let
us
denote
the
absolute
Frobenius
([Kato],
§4)
of
S
log
by
Φ
S
log
:
S
log
→
S
log
.
Let
f
log
:
X
log
→
S
log
be
an
r-pointed
stable
curve
of
genus
g
(as
in
Chapter
I,
Definition
2.1,
so
2g
−
2
+
r
≥
1).
In
general,
we
shall
denote
by
means
of
a
superscript
“F
”
the
result
of
base-changing
by
Φ
S
log
.
Let
def
Φ
X
log
/S
log
:
X
log
→
(X
log
)
F
=
X
log
×
S
log
,Φ
S
log
S
log
be
the
relative
Frobenius.
Deformations
and
FL-Bundles
We
begin
by
reviewing
the
Cartier
isomorphism
(as
in
[Kato],
Theorem
4.12).
Since
a
curve
is
one-dimensional,
this
amounts
to
the
existence
of
an
exact
sequence
of
sheaves
on
X:
log
log
F
→
(Φ
X
log
/S
log
)
−1
(ω
X/S
)
→
0
0
→
(Φ
X
log
/S
log
)
−1
O
X
F
→
O
X
→
ω
X/S
def
where
the
morphism
in
the
middle
is
the
exterior
differentiation
operator
d.
Let
Q
=
log
.
Then
note
that
since
the
above
exact
sequence
is
functorial
with
to
base-
d(O
X
)
⊆
ω
X/S
log
→
S
log
,
the
formation
of
Q
is
likewise
functorial
with
respect
to
base-change.
change
T
We
would
like
to
consider
what
happens
to
this
exact
sequence
when
it
is
tensored
over
(Φ
X
log
/S
log
)
−1
O
X
F
with
(Φ
X
log
/S
log
)
−1
(τ
X
log
/S
log
)
F
.
Let
T
=
(Φ
X
log
/S
log
)
∗
(τ
X
log
/S
log
)
F
.
We
then
obtain
(by
using
the
long
exact
cohomology
sequence
for
higher
direct
images)
the
following
two
exact
sequences
of
sheaves
on
S:
log
0
→
O
S
→
R
1
f
∗
Q
⊗
O
X
T
→
R
1
f
∗
ω
X/S
⊗
O
X
T
0
→
R
1
f
∗
(τ
X
log
/S
log
)
F
→
R
1
f
∗
T
→
R
1
f
∗
Q
⊗
O
X
T
→
0
where
we
use
the
fact
that
f
∗
(Q
⊗
Φ
−1
X
log
/S
log
O
X
F
log
F
τ
X
log
/S
log
)
→
f
∗
(ω
X/S
⊗
O
X
T
)
=
0
by
degree
considerations.
Now
let
us
note
that
T
has
a
natural
logarithmic
connec-
tion
∇
T
obtained
by
declaring
the
sections
of
the
subsheaf
(Φ
X
log
/S
log
)
−1
(τ
X
log
/S
log
)
F
⊆
(Φ
X
log
/S
log
)
∗
(τ
X
log
/S
log
)
F
=
T
to
be
horizontal.
Thus,
by
using
the
above
exact
sequences,
we
can
compute
the
first
de
Rham
cohomology
module
of
T
(where
we
always
understand
T
to
be
equipped
with
the
connection
∇
T
).
58
Proposition
1.1.
We
have
an
exact
sequence:
0
→
R
1
f
∗
(τ
X
log
/S
log
)
F
→
R
1
f
DR,∗
(T
)
→
O
S
→
0
which
is
functorial
with
respect
to
base-change
T
log
→
S
log
.
In
particular,
R
1
f
DR,∗
(T
)
is
a
vector
bundle
of
rank
3g
−
2
+
r
on
S.
Finally,
R
1
f
DR,∗
(T
)
⊆
R
1
f
∗
(T
).
Let
us
denote
by
A
the
R
1
f
∗
(τ
X
log
/S
log
)
F
-torsor
on
S
defined
by
the
above
exact
sequence.
Let
S
log
be
a
fine
log
scheme
whose
underlying
scheme
is
flat
over
Z/p
2
Z,
log
→
S
log
be
an
r-pointed
stable
curve
of
and
such
that
S
log
⊗
Z/pZ
=
S
log
.
Let
X
genus
g
lifting
f
log
.
Then
for
any
r-pointed
stable
curve
Y
log
→
S
log
of
genus
g
that
lifts
(X
log
)
F
→
S
log
,
we
can
associate
a
section
θ
Y
of
A
as
follows.
Consider
the
obstruction
log
→
Y
log
.
to
lifting
the
relative
Frobenius
Φ
X
log
/S
log
:
X
log
→
(X
log
)
F
to
a
morphism
X
This
defines
a
section
θ
Y
of
R
1
f
∗
(T
).
Observe
that
θ
Y
is
independent
of
the
choice
of
lifting
log
.
Indeed,
this
follows
from
the
fact
that
locally,
if
one
changes
the
lifting
X
log
,
the
X
log
obstruction
cocycle
will
change
by
a
derivation
of
X
applied
to
a
function
pulled-back
log
F
via
Φ
X
log
/S
log
from
(X
)
.
But
this
will
always
give
zero.
This
proves
that
θ
Y
depends
only
on
Y
log
.
Let
us
also
observe
that
θ
Y
actually
defines
a
section
of
R
1
f
DR,∗
(T
)
⊆
R
1
f
∗
(T
).
Indeed,
to
see
this,
we
reason
as
follows.
We
work
with
bianalytic
functions,
as
in
Chapter
I,
§1.
Then
the
inverse
image
via
the
relative
Frobenius
of
O
(X
bi
)
F
in
O
X
bi
coincides
with
both
i
L
{(Φ
X
log
/S
log
)
−1
O
X
F
}
and
i
R
{(Φ
X
log
/S
log
)
−1
O
X
F
}.
Thus,
the
two
pull-backs
(from
the
right
and
left)
of
θ
Y
to
O
X
bi
both
correspond
to
the
obstruction
to
lifting
i
L
{(Φ
X
log
/S
log
)
−1
O
X
F
}
=
i
R
{(Φ
X
log
/S
log
)
−1
O
X
F
}
to
a
O
S
-flat
subalgebra
(with
log
structure)
of
O
X
bi
.
This
shows
that
θ
Y
is
horizontal.
log
→
S
log
of
(X
log
)
F
→
S
log
.
Then
Now
suppose
that
we
consider
another
lifting
Z
the
difference
between
the
liftings
Y
log
and
Z
log
naturally
defines
a
section
θ
Y
Z
of
the
vector
bundle
R
1
f
∗
(τ
X
log
/S
log
)
F
.
It
follows
immediately
from
the
definition
of
an
obstruc-
tion
class
that
θ
Y
=
θ
Z
+
θ
Y
Z
.
Thus,
if
there
existed
a
lifting
Y
log
such
that
θ
Y
is
a
section
of
the
subsheaf
R
1
f
∗
(τ
X
log
/S
log
)
F
⊆
R
1
f
DR,∗
(T
),
then
it
would
follow
that
there
exists
another
lifting
Z
log
such
that
θ
Z
=
0.
Thus,
the
relative
Frobenius
would
lift
to
a
log
log
morphism
Ψ
:
X
→
Z
.
But
then
by
pull-back,
Ψ
would
induce
a
nonzero
morphism
log
F
log
of
(Φ
X
log
/S
log
)
∗
(ω
X/S
)
into
ω
X/S
,
which,
by
degree
considerations,
is
absurd.
Thus,
we
1
conclude
that
no
θ
Y
lies
in
R
f
∗
(τ
X
log
/S
log
)
F
⊆
R
1
f
DR,∗
(T
).
In
other
words,
for
every
lifting
Y
log
,
θ
Y
defines
a
section
of
A.
Let
D
be
the
R
1
f
∗
(τ
X
log
/S
log
)
F
-torsor
over
S
of
liftings
of
(X
log
)
F
→
S
log
to
S
log
.
Then,
we
see
that
we
have
defined
a
canonical
morphism
of
R
1
f
∗
(τ
X
log
/S
log
)
F
-torsors
F
:
D→A
Since
any
morphism
of
torsors
is
necessarily
an
isomorphism,
we
see
that
we
have
proven
the
following
result:
([Kato],
Theorem
4.12
(2))
59
Proposition
1.2.
The
canonical
morphism
F
:
D
→
A
is
an
isomorphism.
Let
(E,
∇
E
)
be
a
rank
two
vector
bundle
on
X
with
a
connection
∇
E
(relative
to
f
log
).
Suppose
that
there
exists
a
horizontal
exact
sequence
0
→
T
→
E
→
O
X
→
0
Then
this
exact
sequence
defines
a
section
η
of
R
1
f
DR,∗
(T
)
over
S.
Definition
1.3.
We
shall
call
(E,
∇
E
)
an
FL-(vector)
bundle
if
η
maps
to
O
S
×
⊆
O
S
under
the
map
R
1
f
DR,∗
(T
)
→
O
S
of
Proposition
1.1.
We
shall
call
a
P
1
-bundle
with
connection
(P,
∇
P
)
on
X
log
an
FL-(P
1
)-bundle
if
it
can
be
written
étale
locally
on
S
as
the
P
1
-bundle
associated
to
an
FL-vector
bundle.
Remark.
The
letters
“FL”
stand
for
Frobenius
lifting.
Since
a
FL-bundle
defines
a
section
of
the
torsor
A,
it
follows
by
Proposition
1.2
that
it
also
defines
a
lifting
Y
log
→
S
log
of
(X
log
)
F
→
S
log
.
Also,
we
shall
see
below
(Corollary
1.5)
that,
at
least
if
S
is
reduced,
then
if
a
horizontal
exact
sequence
as
above
exists,
it
is
necessarily
unique.
The
p-Curvature
of
an
FL-Bundle
Let
us
assume
for
the
rest
of
the
Section
that
p
is
odd.
Throughout
this
Chapter
the
notion
of
the
p-curvature
of
a
bundle
with
connection
in
characteristic
p
will
play
an
important
role.
We
refer
to
[Katz],
§5,
6;
[Ogus],
§1.2,
1.3
for
basic
facts
concerning
the
p-curvature.
([Katz],
of
course,
does
not
handle
the
arbitrary
“log-smooth”
case,
but
the
definitions
and
proofs
(of
the
properties
that
we
will
need)
go
through
without
change.
At
any
rate,
on
the
sorts
of
curves
that
we
are
working
with,
the
theory
of
[Katz],
§5,
6,
is
literally
valid
on
an
open,
schematically
dense
subset,
and
many
assertions
can
be
checked
after
restriction
to
such
an
open
subset.)
Let
(E,
∇
E
)
be
an
FL-bundle.
We
would
like
to
compute
the
p-curvature
of
(E,
∇
E
).
The
p-curvature
P
will
be
a
horizontal
section
of
T
∨
⊗
O
X
Ad(E).
Occasionally,
we
shall
think
of
P
as
a
morphism
Ad(E)
→
T
∨
or
a
morphism
T
→
Ad(E)
(using
the
fact
that
Ad(E)
is
self-dual).
By
abuse
of
notation,
we
shall
also
refer
to
these
morphisms
by
the
notation
P.
Now,
first
of
all,
since
∇
E
stabilizes
the
filtration
T
⊆
E,
by
functoriality,
P
also
respects
this
filtration.
Secondly,
since
T
and
O
X
clearly
have
p-curvature
zero,
P
not
only
respects
the
filtration,
but
is
nilpotent,
i.e.,
P
:
T
→
Ad(E)
maps
into
the
subbundle
T
⊆
Ad(E)
(given
by
endomorphisms
of
E
obtained
by
projecting
E
→
O
X
,
mapping
O
X
to
T
,
and
then
injecting
T
→
E).
Thus,
P
basically
amounts
to
a
morphism
from
T
to
T
,
i.e.,
a
section
of
f
∗
O
X
=
O
S
.
Proposition
1.4.
Assume
that
p
is
odd.
If
(E,
∇
E
)
is
an
FL-bundle
on
X
log
,
then
P
:
T
→
Ad(E)
is
given
by
multiplication
T
→
T
by
−1,
followed
by
the
inclusion
T
→
Ad(E).
60
Proof.
Since
the
universal
example
of
an
FL-bundle
on
an
r-pointed
stable
curve
of
genus
g
is
given
by
the
torsor
A
over
M
g,r
,
which
is
smooth,
it
suffices
to
check
the
assertion
after
restriction
to
a
closed
point
of
this
universal
A.
Thus,
we
can
assume
that
S
=
Spec(k),
where
k
is
a
finite
field.
Then
S
has
a
canonical
lifting
to
a
flat
scheme
over
Z/p
2
Z,
namely,
S
=
Spec(W
(k)/p
2
W
(k))
(where
W
(k)
is
the
ring
of
Witt
vectors
with
coefficients
in
k),
→
S,
we
may
and
S
has
a
natural
Frobenius
lifting
Φ
S
.
Thus,
for
any
smooth
scheme
U
,
that
is,
a
S-linear
→
U
F
speak
of
a
Frobenius
lifting
(mod
p
2
)
on
U
morphism
Φ
:
U
def
whose
reduction
modulo
p
is
the
relative
Frobenius
U
→
U
F
of
U
=
U
⊗
Z/pZ.
Now
let
us
take
U
to
be
an
affine
open
subset
U
⊆
X,
at
which
f
is
smooth,
and
→
S
be
a
smooth
lifting
of
U
,
and
let
t
be
a
which
contains
no
marked
points.
Let
U
local
coordinate
on
U
.
By
the
interpretation
of
FL-bundles
in
terms
of
obstructions
to
Frobenius
liftings,
we
may
compute
E
by
using
as
follows:
Over
U
,
E|
U
∼
=
T
|
U
⊕
O
U
.
Let
us
write
sections
of
E
relative
to
the
decomposition
T
|
U
⊕
O
U
and
the
basis
given
by
(Φ
−1
(
d
)
F
,
0);
(0,
1)
;
and
let
us
denote
by
∇
the
connection
∇
E
applied
in
the
X
log
/S
log
dt
d
.
Then
∇
is
given
by
adding
to
the
direct
sum
connection
the
matrix
direction
dt
0
1
p
Φ
0
0
where
the
map
Φ
is
the
derivative
(with
respect
to
t)
of
some
local
Frobenius
lifing
.
Since
t
F
→
(1
+
t)
p
−
1
is
a
Frobenius
lifting,
Φ(t
F
)
must
be
of
the
form
Φ
on
U
(1+t)
p
−1+f
(t),
for
some
function
f
(t)
on
U
.
Therefore,
p
1
Φ
is
of
the
form
(1+t)
p−1
+f
(t).
This
gives
∇(1,
0)
=
0;
∇(0,
1)
=
((1
+
t)
p−1
+
f
,
0).
Therefore,
(∇)
p
(1,
0)
=
0
and
d
p
∇
p
(0,
1)
=
(p
−
1)!(1,
0)
(since
(
dt
)
f
=
0).
Finally,
it
follows
easily
from
using
the
fact
×
that
F
p
is
a
cyclic
group
that
(p
−
1)!
=
−1
(in
F
p
).
This
completes
the
proof.
Corollary
1.5.
Assume
that
p
is
odd.
Let
(E,
∇
E
)
be
a
rank
two
vector
bundle
with
logarithmic
connection
on
X
log
(over
S
log
)
defined
by
a
section
η
of
R
1
f
DR,∗
(T
).
Then
(E,
∇
E
)
is
an
FL-bundle
if
and
only
if
its
p-curvature
is
nonzero
at
some
point
of
every
fiber
of
f
:
X
→
S.
Proof.
The
“only
if”
part
follows
from
Proposition
1.4.
On
the
other
hand,
suppose
that
the
image
of
η
under
the
map
R
1
f
DR,∗
(T
)
→
O
S
of
Proposition
1.1
vanishes
at
a
point.
By
restricting
to
that
point,
we
may
assume
that
S
is
the
spectrum
of
a
field,
and
that
the
image
of
η
in
O
S
vanishes
identically.
But
then,
it
follows
from
the
exact
sequence
of
Proposition
1.1
that
(E,
∇
E
)
is
the
pull-back
of
a
bundle
under
Frobenius.
Then
its
p-curvature
must
vanish
identically,
which
contradicts
our
assumption.
Corollary
1.6.
Assume
that
p
is
odd
and
that
S
is
reduced.
Let
(E,
∇
E
)
be
an
FL-bundle.
Let
U
⊆
X
be
an
open
subset;
(L,
∇
L
)
be
a
line
bundle
with
logarithmic
connection
on
U
log
;
61
and
ι
:
L
→
E|
U
be
a
horizontal
morphism
of
O
U
-modules
with
logarithmic
connections.
Then
ι
factors
through
the
injection
T
|
U
→
E|
U
in
the
definition
of
E
as
an
FL-bundle.
Proof.
By
shrinking
U
,
we
may
assume
that
f
:
X
→
S
is
smooth
on
U
,
and
that
U
stays
away
from
the
marked
points.
We
may
also
assume
that
the
composite
of
ι
with
the
projection
E|
U
→
O
U
is
an
isomorphism.
Thus,
we
obtain
a
horizontal
isomorphism
of
line
bundles
L
→
O
U
.
But
this
implies
that
(L,
∇
L
)
has
p-curvature
zero.
Thus,
we
get
a
horizontal
isomorphism
L
⊕
T
|
U
→
E|
U
.
Since
the
left-hand
side
has
p-curvature
zero,
the
same
is
true
of
the
right-hand
side.
But
this
contradicts
Proposition
1.4.
Remark.
It
is
not
difficult
to
construct
counterexamples
to
Corollary
1.6
if
one
does
not
assume
that
S
is
reduced.
§2.
The
Verschiebung
on
Indigenous
Bundles
In
this
Section,
we
define
a
“Verschiebung”
morphism
on
the
space
S
g,r
of
Schwarz
structures
that
takes
values
in
the
space
of
square
differentials
(twisted
by
Frobenius).
We
then
prove
various
basic
properties
of
this
morphism,
such
as
computing
its
derivative.
This
computation
reveals
that
the
derivative
looks
rather
like
the
Verschiebung
morphism
for
the
Jacobian
of
the
curve,
thus
justifying
the
terminology.
On
the
other
hand,
as
we
shall
see
in
§3,
unlike
the
Verschiebung
of
the
Jacobian
which
only
pertains
to
H
1
of
the
curve,
the
Verschiebung
on
indigenous
bundles
pertains
to
a
nonabelian
invariant
of
the
curve,
namely,
the
nilpotent
indigenous
bundles
on
the
curve.
It
turns
out
that
the
study
of
nilpotent
indigenous
bundles,
and
thus
of
the
Verschiebung
on
indigenous
bundles
are
central
to
understanding
uniformization
theory
in
the
p-adic
context.
In
this
Section,
M
g,r
(respectively,
S
g,r
)
will
denote
the
moduli
stack
of
r-pointed
curves
of
genus
g
(respectively,
equipped
with
a
Schwarz
structure)
over
F
p
.
We
assume
throughout
this
Section
that
p
is
odd.
The
Definition
of
the
Verschiebung
Let
S
log
be
a
fine
noetherian
log
scheme
over
F
p
.
Let
f
log
:
X
log
→
S
log
be
an
r-
pointed
stable
curve
of
genus
g.
Let
D
⊆
X
be
the
divisor
of
marked
points.
Let
(E,
∇
E
)
be
an
indigenous
bundle
on
X
log
(see
§2
of
Chapter
I
for
more
on
such
bundles).
We
remark
here
that
throughout
this
paper,
when
we
do
various
manipulations
with
indigenous
bundles,
it
will
be
simpler
to
work
with
vector
bundles,
rather
than
P
1
-bundles.
Of
course,
indigenous
vector
bundles
only
exist
under
certain
conditions
(cf.
Proposition
2.6
of
Chap-
ter
I),
but
this
will
not
pose
any
problem,
since
we
can
always
either
Zariski
localize
on
the
curve,
or
pass
to
some
sort
of
covering
of
the
curve,
and
then
descend
for
the
final
result.
Thus,
in
the
future,
for
the
rest
of
the
paper,
we
shall
act
as
if
this
problem
does
not
exist,
and
always
work
with
indigenous
vector
bundles,
when
it
is
technically
more
comfortable
to
do
so.
62
We
maintain
the
notation
of
§1
for
the
various
Frobenius
morphisms
and
for
T
=
(Φ
X
log
/S
log
)
∗
(τ
X
log
/S
log
)
F
.
Let
P
E
:
T
→
Ad(E)
be
the
p-curvature
of
(E,
∇
E
).
Consider
the
composite
of
P
E
with
its
dual
P
E
∨
.
This
composite
is
a
horizontal
morphism
T
→
T
∨
,
log
⊗2
F
hence
defines
a
section
of
(f
∗
(ω
X/S
)
)
.
Let
V
E
be
−
12
times
this
section.
Another
way
to
put
the
definition
of
V
E
is
as
follows:
We
consider
the
square
(P
E
)
2
:
(T
)
⊗2
→
End(E)
of
P
E
,
take
the
trace,
and
multiply
by
−
12
.
Yet
another
way
to
put
the
definition
of
V
E
is
that
it
is
the
determinant
of
P
E
(regarded
as
a
map
(T
)
⊗2
→
O
X
).
Proposition
2.1.
Assume
that
S
is
reduced.
Then
V
E
is
zero
if
and
only
if
the
image
of
P
E
consists
of
nilpotent
endomorphisms
of
E.
Proof.
Immediate
from
the
definitions.
Thus,
we
may
think
of
V
E
as
a
measure
of
how
nilpotent
P
E
is.
Note,
in
particular,
that
at
a
marked
point,
by
definition
∇
E
has
nilpotent
monodromy,
so
the
p-curvature
is
already
nilpotent
there.
Thus,
V
E
has
zeroes
at
all
the
marked
points.
By
abuse
of
notation,
we
log
⊗2
)
(−D))
F
.
shall
denote
by
V
E
the
resulting
section
of
(f
∗
(ω
X/S
log
⊗2
Definition
2.2.
We
shall
refer
to
the
section
V
E
of
(f
∗
(ω
X/S
)
(−D))
F
as
the
Ver-
schiebung
of
the
indigenous
bundle
(E,
∇
E
).
log
Thus,
in
the
universal
case,
when
f
log
:
C
log
→
M
g,r
is
the
universal
r-pointed
stable
curve
of
genus
g,
we
obtain
a
morphism
of
M
g,r
-schemes:
V
g,r
:
S
g,r
→
Q
g,r
where
Q
g,r
is
the
geometric
vector
bundle
corresponding
to
(i.e.,
Spec
of
the
symmetric
log
⊗2
)
(−D))
F
.
Note
that
both
S
g,r
and
Q
g,r
are
of
dimension
algebra
of
the
dual
of)
(f
∗
(ω
X/S
3g−3+r
over
M
g,r
.
The
rest
of
this
Chapter
will
be
devoted
to
studying
this
Verschiebung
morphism
V
g,r
.
Let
us
begin
with
some
observations
concerning
the
degree
of
V
g,r
as
a
polynomial
map.
Let
∇
E
=
∇
E
+
θ
be
a
logarithmic
connection
on
E
that
makes
it
an
indigenous
bundle;
here
we
may
assume
that
θ
is
an
Ad(E)-valued
differential
which
defines
a
square-
nilpotent
endomorphism
of
E
and
which
corresponds
to
a
square
differential
θ
SD
.
Thus,
θ
2
=
0.
If
we
then
compute
the
p-curvature
of
(E,
∇
E
)
by,
say,
working
locally
on
U
⊆
X
where
there
is
a
local
coordinate
x,
and
letting
∇
(respectively,
∇
;
θ
x
)
denote
∇
E
def
d
(respectively,
∇
E
;
θ)
applied
in
the
direction
∂
=
dx
,
we
note
that
because
θ
x
2
=
0,
all
the
terms
that
involve
θ
x
more
than
12
(p
+
1)
times
must
vanish.
Moreover,
there
is
only
one
term
that
involves
θ
x
exactly
12
(p
+
1)
times,
namely:
63
θ
x
∇
θ
x
∇
.
.
.
∇
θ
x
that
is,
alternating
θ
x
’s
and
∇’s,
with
a
total
of
12
(p
+
1)
copies
of
θ
x
and
12
(p
−
1)
copies
of
∇.
For
future
reference
let
us
call
this
term
ξ.
Note
that
since
θ
x
2
=
0,
any
string
θ
x
∇
θ
x
can
be
rewritten
θ
x
[∇,
θ
x
]
(where
the
brackets
denote
the
commutator),
and
that
this
commutator
[∇,
θ
x
]
is
a
linear
operator
(that
is,
linear
over
O
X
).
Moreover,
this
linear
operator
[∇,
θ
x
]
preserves
the
Hodge
filtration
of
E.
Thus,
ξ
may
be
rewritten
as
θ
x
times
[∇,
θ
x
]
to
some
power.
Since
[∇,
θ
x
]
preserves
the
Hodge
filtration,
it
thus
follows
that
ξ
is
a
linear
operator
on
E
which
is
nilpotent
with
respect
to
the
Hodge
filtration.
In
particular,
ξ
2
=
0.
At
any
rate,
we
may
at
least
conclude
that
in
the
expression
for
the
trace
of
the
square
of
the
p-curvature
of
∇
E
,
θ
occurs
no
more
than
p
times
in
each
term.
We
thus
obtain
the
following
result:
(*)
Relative
to
the
affine
structures
of
S
g,r
and
Q
g,r
,
the
pull-backs
of
the
affine
variables
on
Q
g,r
via
the
morphism
V
g,r
are
polynomials
in
the
affine
variables
of
S
g,r
of
degree
≤
p.
In
fact,
we
would
like
to
conclude
a
stronger
result,
namely
that
the
degree
is
exactly
p.
In
order
to
do
this,
we
need
to
enlist
the
aid
of
Jacobson’s
formula
(see,
e.g.,
[Jac],
pp.
186-187):
This
formula
states
that
if
a
and
b
are
elements
of
an
associative
ring
R
of
characteristic
p,
then
p
p
p
(a
+
b)
=
a
+
b
+
p−1
s
i
(a,
b)
i=1
where
the
s
i
(a,
b)
are
given
by
the
formula:
p−1
(ad(ta
+
b))
(a)
=
p−1
is
i
(a,
b)t
i−1
i=1
computed
in
the
ring
R[t],
where
t
is
an
indeterminate.
In
our
case,
we
let
b
=
∇
and
a
=
θ
x
,
and
we
wish
to
compute
the
s
j
(a,
b),
where
j
=
12
(p
−
1).
Let
η
be
the
coefficient
multiplying
t
j−1
in
the
expression
(ad(ta
+
b))
p−1
(a).
Let
α
=
ad(a);
β
=
ad(b).
Then
the
terms
in
η
look
like
12
(p
−
3)
copies
of
α
and
12
(p
+
1)
copies
of
β
applied
to
a
in
some
order.
Now
ultimately,
in
order
to
compute
the
Verschiebung,
we
are
interested
in
computing
tr(η
·
ξ).
Let
τ
be
one
of
the
terms
that
make
up
η.
Now
we
separate
the
analysis
of
τ
into
two
cases:
the
case
where
τ
begins
with
an
α,
and
the
case
where
τ
begins
with
a
β.
Suppose
that
τ
begins
with
an
α.
Thus,
τ
looks
like
α
γ
1
.
.
.
γ
p−2
a
64
where
each
γ
i
is
either
α
or
β.
Now
let
us
note
that
when
one
applies
α
=
ad(θ
x
)
or
β
=
ad(∇)
to
a
linear
operator
on
E,
one
gets
back
a
linear
operator.
Moreover,
α
applied
to
any
linear
operator
on
E
yields
a
linear
operator
that
preserves
the
Hodge
filtration
on
E.
Thus,
we
conclude
that
τ
is
a
linear
operator
on
E
that
preserves
the
Hodge
filtration.
Since
ξ
is
nilpotent
with
respect
to
the
Hodge
filtration,
it
thus
follows
that
tr(τ
·
ξ)
=
0.
Now
suppose
that
τ
begins
with
a
β:
β
γ
1
.
.
.
γ
p−2
a
Let
us
denote
by
σ
the
linear
operator
on
E
given
by
γ
1
.
.
.
γ
p−2
a
(i.e.,
we
leave
off
the
initial
β).
We
would
like
to
show
that
σ
preserves
the
Hodge
filtration
on
E.
To
see
this,
first
note
that
among
the
γ
i
’s,
the
number
of
β’s
is
exactly
one
greater
than
the
number
of
α’s.
Also,
note
that
(by
Griffiths
transversality)
relative
to
the
Hodge
filtration
on
Ad(E),
applying
β
decreases
the
filtration
index
l
(in
F
l
(Ad(E)))
by
at
most
one,
while
applying
α
always
increases
the
filtration
index
l
by
one.
Thus,
since
a
=
θ
x
∈
F
1
(Ad(E)),
it
follows
that
σ
∈
F
0
(Ad(E)),
i.e.,
it
fixes
the
Hodge
filtration
on
E,
and
so
σ
·
ξ
is
nilpotent
with
respect
to
the
Hodge
filtration.
In
particular,
tr(σ
·
ξ)
=
0.
Since
the
trace
map
is
horizontal,
we
thus
obtain
that
tr([∇,
σ
·
ξ])
=
0.
Therefore,
when
we
multiply
τ
=
β(σ)
by
ξ
and
take
the
trace,
we
get
tr(τ
·
ξ)
=
tr([∇,
σ
·
ξ])
−
tr(σ
·
[∇,
ξ])
=
−tr(σ
·
[∇,
ξ])
In
other
words,
tr(τ
·
ξ)
depends
only
on
the
images
of
σ
and
[∇,
ξ]
in
F
0
/F
1
(Ad(E)).
To
compute
the
image
of
σ
in
F
0
/F
1
(Ad(E)),
we
must
analyze
σ
in
greater
detail.
Now
we
saw
that
when
we
compute
σ
by
applying
α’s
and
β’s
to
a,
β
decreases
the
filtration
index
by
at
most
one.
The
only
time
it
fails
to
decrease
the
filtration
index
by
one
is
when
it
is
applied
to
a
linear
operator
which
already
has
a
nontrivial
image
in
F
−1
/F
0
(Ad(E)).
If
this
should
occur
even
once,
then
the
net
change
in
the
filtration
index
as
a
result
of
applying
all
the
γ
i
’s
(in
the
computation
of
σ)
to
a
is
≤
0.
Thus,
if
this
occurs
even
once,
σ
∈
F
1
(Ad(E)),
so
tr(σ
·
[∇,
ξ])
=
0.
On
the
other
hand,
if,
during
the
calculation
of
σ,
we
apply
α
to
a
linear
operator
in
F
1
(Ad(E)),
we
get
zero.
Let
us
call
the
case
where
neither
of
these
two
phenomena
ever
occurs
the
nondegenerate
case.
Thus,
only
the
nondegenerate
terms
τ
will
make
a
nonzero
contribution.
Let
us
suppose
that
τ
is
nondegenerate.
Then
in
order
to
compute
the
image
of
σ
in
F
0
/F
1
(Ad(E)),
it
suffices
to
merely
keep
track
of
the
leading
term
(relative
to
the
Hodge
filtration
on
Ad(E))
as
we
apply
the
various
γ
i
’s.
Now
let
us
note
that
it
follows
from
the
fact
that
the
Kodaira-Spencer
morphism
for
the
Hodge
filtration
on
E
is
the
identity
that
if
l
≥
1,
then
applying
β
to
a
linear
operator
L
in
F
l
(Ad(E))
yields
a
linear
operator
in
F
l−1
(Ad(E))
whose
image
in
F
l−1
/F
l
(Ad(E))
is
the
image
of
L
in
F
l
/F
l+1
(Ad(E))
times
∂.
On
the
other
hand,
if
l
≤
1,
then
applying
α
to
a
linear
operator
L
in
F
l
Ad(E)
yields
a
linear
operator
in
F
l+1
(Ad(E))
whose
image
in
F
l+1
/F
l+2
(Ad(E))
is
the
image
65
of
L
in
F
l
/F
l+1
(Ad(E))
times
θ
SD
·
∂.
Thus,
if
τ
is
nondegenerate,
then
its
contribution
−tr(σ
·
[∇,
ξ])
is
given
by
(θ
SD
·
∂)
p
·
(∂)
p
.
It
remains
to
compute
the
number
of
nondegenerate
terms
τ
.
Let
us
call
this
number
N
p
,
and
regard
it
as
an
element
of
F
p
.
We
are
interested
in
whether
or
not
N
p
∈
F
p
is
zero.
Although
one
can
presumably
compute
N
p
explicitly
using
some
sort
of
combinatorial
argument,
we
prefer
to
take
the
following
approach.
Note
that
N
p
does
not
depend
on
g
or
r
or
on
the
particular
curve
f
log
:
X
log
→
S
log
,
but
only
on
p.
Thus,
it
suffices
to
show
that
(for
each
odd
prime
p)
N
p
=
0
for
one
particular
curve
(with
g
and
r
arbitrary).
We
shall
do
this
in
§3
when
we
do
various
computations
with
elliptic
curves.
Thus,
we
log
⊗2
)
(−D))
F
),
when
applied
to
(∂
F
)
2
yields
obtain
that
V
E
(which
is
a
section
of
(f
∗
(ω
X/S
the
function
c
·
(θ
SD
·
∂)
p
·
(∂)
p
,
where
c
∈
F
×
p
.
Put
another
way,
the
degree
p
component
of
V
g,r
is
a
morphism:
[p]
log
⊗2
log
⊗2
V
g,r
:
{(f
∗
(ω
X/S
)
(−D))
∨
}
F
→
S
p
(f
∗
(ω
X/S
)
(−D))
∨
which
is
equal
to
c
times
the
Frobenius
morphism,
i.e.,
the
p
th
-power
map.
We
shall
see
later
in
our
computations
with
elliptic
curves
that
c
=
−1.
Thus,
we
thus
obtain
the
following
important
result:
Theorem
2.3.
Relative
to
the
affine
structures
of
S
g,r
and
Q
g,r
,
the
pull-backs
of
the
affine
variables
on
Q
g,r
via
the
morphism
V
g,r
are
polynomials
in
the
affine
variables
of
[p]
S
g,r
of
degree
exactly
p,
with
the
leading
term
V
g,r
(i.e.,
the
degree
p
component)
given
by
−1
times
the
p
th
-power
map.
In
particular,
V
g,r
is
a
finite,
flat
morphism
of
degree
p
3g−3+r
.
Proof.
It
remains
to
verify
the
last
assertion
(that
V
g,r
is
finite
and
flat
of
the
right
degree).
Let
U
=
Spec(A)
→
M
g,r
be
étale.
Then
over
U
,
we
may
choose
affine
coordinates
X
i
and
Y
j
of
S
g,r
and
Q
g,r
so
that
V
g,r
looks
like
(the
map
induced
on
Spec’s
by):
def
def
B
=
A[Y
1
,
.
.
.
,
Y
3g−3+r
]
→
C
=
A[X
1
,
.
.
.
,
X
3g−3+r
]
where
Y
i
→
f
i
(X
1
,
.
.
.
,
X
3g−3+r
),
and
f
i
is
of
the
form
“−X
i
p
plus
terms
of
lower
degree.”
Then
it
is
easy
to
see
that,
as
a
B-module,
C
is
generated
by
monomials
of
the
form
3g−3+r
X
i
e
i
i=1
where
0
≤
e
i
≤
p
−
1.
In
particular,
C
is
a
finite
B-module,
so
V
g,r
is
finite.
Since
V
g,r
is
a
finite
morphism
between
regular
algebraic
stacks
of
the
same
dimension,
it
follows
from
commutative
algebra
that
V
g,r
is
flat.
To
compute
the
degree
of
V
g,r
,
let
P
=
Proj
A
(C[T
])
66
(with
the
grading
such
that
T
and
the
X
i
’s
have
degree
one).
Then
Spec(C)
⊆
P
is
an
affine
open
subset.
Let
S
i
⊆
P
be
the
hypersurface
which
is
the
closure
of
the
zero
locus
of
f
i
.
Then
the
scheme-theoretic
intersection
V
of
all
the
S
i
’s
has
degree
p
3g−3+r
over
Spec(A).
Also,
the
intersection
of
V
with
P
−
Spec(C)
=
V
+
(T
)
=
Proj(C)
is
just
p
)
⊆
Proj(C),
which
is
the
empty
set.
Thus,
V
⊆
Spec(C),
and
so
V
+
(X
1
p
,
.
.
.
,
X
3g−3+r
deg(V
g,r
)
=
deg(V
→
Spec(A))
=
p
3g−3+r
.
Before
continuing,
we
introduce
some
more
terminology
that
will
be
of
use
in
the
future:
Definition
2.4.
Let
(E,
∇
E
)
be
an
indigenous
bundle
on
X
log
.
Then,
we
will
say
that
(E,
∇
E
)
is
nilpotent
if
V
E
is
zero.
We
will
say
that
(E,
∇
E
)
is
admissible
if
P
E
∨
:
Ad(E)
→
T
∨
is
surjective.
Note
that
the
terminology
of
a
“nilpotent
indigenous
bundle”
that
we
have
introduced
here
is
(by
Proposition
2.1)
consistent
with
that
of
[Katz].
Also,
let
us
observe
that
the
nilpotent
bundles
form
a
closed
subscheme
N
g,r
⊆
S
g,r
while
the
admissible
bundles
form
an
open
subscheme
adm
S
g,r
⊆
S
g,r
adm
adm
We
shall
see
later
in
this
Chapter
that
S
g,r
N
g,r
is
nonempty
and
that
neither
S
g,r
nor
N
g,r
is
contained
in
the
other.
Note,
further,
that
Theorem
2.3
implies
that
the
natural
morphism
N
g,r
→
M
g,r
is
finite
and
flat
of
degree
p
3g−3+r
.
Finally,
we
observe
that
one
thing
which
is
interesting
about
nilpotent
indigenous
bundles
is
that
(if
S
is
the
spectrum
of
a
perfect
field,
then)
by
a
result
of
[Falt]
such
indigenous
bundles
arise
as
the
crystalline
Dieudonné
modules
of
certain
finite,
flat
group
schemes
on
X
log
.
This
point
of
view
will
be
pursued
further
in
later
chapters.
This
observation
is
the
main
reason
for
studying
N
g,r
and
V
g,r
,
which
is
the
goal
of
the
present
Chapter.
67
The
p-Curvature
of
an
Admissible
Indigenous
Bundle
As
a
prélude
to
finding
out
more
about
V
g,r
,
it
is
worth
looking
at
various
basic
properties
of
the
p-curvature
of
an
admissible
indigenous
bundle.
Proposition
2.5.
There
is
a
bijective
correspondence
between
nilpotent,
admissible
in-
digenous
bundles
(up
to
tensor
product
with
a
line
bundle
of
order
two)
and
FL-bundles
whose
projectivizations
are
indigenous
given
as
follows:
If
(E,
∇
E
)
is
a
nilpotent,
admissi-
ble
indigenous
bundle
on
X
log
,
let
P
E
:
T
→
Ad(E)
be
its
p-curvature.
Then
the
kernel
of
P
E
∨
:
Ad(E)
→
T
∨
is
an
FL-bundle.
Moreover,
if
(E,
∇
E
)
is
a
nilpotent,
admissible
indigenous
bundle
on
X
log
,
there
exists
a
unique
rank
one
subbundle
M
⊆
E
that
is
annihilated
by
the
endomorphisms
in
the
image
of
P
E
.
This
subbundle
M
is
stabilized
by
∇
E
.
The
induced
connection
∇
M
has
p-curvature
zero,
and
we
have
a
horizontal
isomorphism
M
⊗2
∼
=
T
.
Finally,
suppose
that
S
is
reduced,
and
(E,
∇
E
)
is
nilpotent,
admissible,
and
indige-
nous.
Let
U
⊆
X
be
an
open
set,
and
(L,
∇
L
)
a
line
bundle
with
logarithmic
connection
(with
respect
to
f
log
:
X
log
→
S
log
)
on
U
.
Let
ι
:
L
→
E|
U
be
a
horizontal
morphism.
Then
ι
factors
through
M
⊆
E.
Proof.
First
of
all,
since
(E,
∇
E
)
is
admissible,
P
E
∨
is
surjective,
and
thus
its
kernel
is
a
rank
two
vector
bundle
F
which
is
stabilized
by
∇
E
,
hence
gets
a
connection
∇
F
.
Since
(E,
∇
E
)
is
nilpotent,
it
follows
that
T
∼
=
Im(P
E
)
⊆
F,
and
that
this
inclusion
T
→
F
is
locally
split.
Also,
this
inclusion
T
→
F
is
necessarily
horizontal,
and
we
also
have
a
horizontal
isomorphism
F/T
∼
=
O
X
.
Thus,
in
order
to
show
that
(F,
∇
F
)
is
an
FL-bundle,
it
suffices
to
show
that
the
p-curvature
of
(F,
∇
F
)
is
nonzero
generically
on
every
fiber
of
f
:
X
→
S
(by
Corollary
1.5).
Thus,
we
may
assume
that
S
is
the
spectrum
of
a
field.
Now
on
some
nonempty
open
set
U
⊆
X,
there
is
line
bundle
with
connection
(L,
∇
L
)
and
a
horizontal
surjection
μ
:
E
→
L
such
that
Im(P
E
|
U
)
is
given
by
endomorphisms
that
vanish
on
Ker(μ)
and
whose
image
is
in
Ker(μ).
Then
sorting
through
what
we
have
done,
we
see
that
we
have
a
horizontal
isomorphism
F|
U
∼
=
E|
U
⊗
L
−1
.
Since
L
has
p-curvature
zero,
the
fact
that
E|
U
has
nonzero
p-curvature
implies
that
the
same
is
true
of
F|
U
.
This
completes
the
proof
that
(F,
∇
F
)
is
an
FL-bundle.
On
the
other
hand,
if
we
are
given
an
FL-bundle
(F,
∇
F
)
whose
projectivization
(P,
∇
P
)
is
indigenous,
let
(E,
∇
E
)
be
an
indigenous
vector
bundle
whose
projectivization
is
(P,
∇
P
).
Then
there
exists
a
line
bundle
with
connection
(L,
∇
L
)
(relative
to
f
log
:
X
log
→
S
log
)
such
that
we
have
a
horizontal
isomorphism
F
⊗
O
X
L
∼
=
E.
Taking
determinants,
∨
T
we
thus
get
a
horizontal
isomorphism
L
⊗2
∼
,
so
the
p-curvature
of
L
must
be
zero.
=
Thus,
under
the
natural
identification
of
Ad(F)
with
Ad(E),
we
see
that
the
p-curvatures
of
(E,
∇
E
)
and
(F,
∇
F
)
coincide.
Thus,
by
our
computation
in
Proposition
1.4,
(E,
∇
E
)
is
admissible
and
nilpotent.
Also,
it
is
easy
to
see
that
these
two
procedures
are
inverse
to
one
another,
thus
proving
the
bijective
correspondence.
We
take
M
⊆
E
to
be
T
⊗
L
⊆
68
F
⊗L∼
=
E.
The
remaining
assertions
follow
immediately
from
what
we
have
done
so
far,
plus
Proposition
1.6.
Proposition
2.6.
Let
(E,
∇
E
)
be
an
indigenous
bundle
on
X
log
.
Let
P
E
:
T
→
Ad(E)
be
its
p-curvature.
Then:
(1)
We
shall
call
the
composite
H
E
:
T
→
τ
X
log
/S
log
of
P
E
with
the
pro-
jection
E
→
τ
X
log
/S
log
arising
from
the
Hodge
filtration
on
E
the
square
Hasse
invariant
of
(E,
∇
E
).
If
(E,
∇
E
)
is
admissible,
then
H
E
is
nonzero.
(2)
Suppose
that
(E,
∇
E
)
is
admissible.
Then
the
zero
locus
V
(H
E
)
⊆
X
is
a
divisor
D
E
which
is
finite,
flat,
and
of
degree
(p
−
1)(2g
−
2
+
r)
over
S.
We
shall
call
it
the
double
supersingular
divisor
of
(E,
∇
E
).
(3)
Suppose
that
the
indigenous
bundle
(E,
∇
E
)
is
admissible
and
nilpotent.
Then
there
exists
a
line
bundle
H
on
X
whose
square
H
⊗2
is
isomorphic
to
T
∨
⊗
τ
X
log
/S
log
,
together
with
a
section
χ
of
H
over
X
whose
square
is
H
E
.
We
shall
call
χ
the
Hasse
invariant
of
(E,
∇
E
).
In
particular,
there
exists
a
divisor
E
E
⊆
X
such
that
D
E
=
2
E
E
.
We
shall
call
E
E
the
supersingular
divisor
of
(E,
∇
E
).
(4)
Suppose
that
S
is
reduced.
Then
any
two
nilpotent,
admissible
indige-
nous
bundles
with
the
same
supersingular
locus
are
isomorphic.
Proof.
For
(1),
it
suffices
to
prove
the
statement
after
we
restrict
to
a
fiber
of
f
:
X
→
S;
thus,
we
may
assume
that
S
is
the
spectrum
of
a
field.
If
H
E
were
zero,
then
that
would
mean
that
Im(P
E
)
lands
in
F
0
(Ad(E)).
Now
F
0
(Ad(E))
surjects
onto
O
X
.
If
the
image
of
P
E
in
O
X
is
nontrivial,
we
get
a
contradication
as
follows:
On
the
one
hand,
Im(P
E
)
is
stabilized
by
∇
E
.
On
the
other
hand,
the
fact
that
the
Kodaira-Spencer
morphism
of
the
Hodge
filtration
is
an
isomorphism
means
that
∇
E
applied
to
Im(P
E
)
will
not
be
in
log
F
0
(Ad(E)).
If
the
image
of
P
E
in
O
X
is
trivial,
then
it
must
lie
in
F
1
(Ad(E))
∼
=
ω
X/S
.
Then,
by
using
the
fact
that
the
Kodaira-Spencer
morphism
is
an
isomorphism,
we
again
get
a
contradiction.
Assertion
(2)
follows
immediately
from
(1).
Now
suppose
that
(E,
∇
E
)
is
nilpotent,
admissible,
and
indigenous.
Let
M
⊆
E
be
the
rank
one
subbundle
of
Proposition
2.5.
Let
E
→
N
be
the
surjection
arising
from
the
Hodge
filtration.
Then
composing
the
injection
M
→
E
with
this
surjection,
we
get
a
morphism
χ
:
M
→
N
,
whose
square
(under
the
identifications
M
⊗2
∼
=
T
;
N
⊗2
∼
=
τ
X
log
/S
log
)
is
equal
to
the
Hasse
invariant.
Thus,
if
we
let
E
E
be
the
zero
locus
of
χ,
we
have
D
E
=
2
E
E
.
This
proves
(3).
To
prove
(4),
we
assume
that
S
is
the
spectrum
of
a
field,
and
that
we
have
two
connections
∇
E
and
∇
E
on
E,
both
of
which
make
E
a
nilpotent,
admissible
indigenous
bundle,
and
such
that
the
respective
supersingular
divisors
E
E
and
E
E
coincide.
Let
us
also
69
assume
that
∇
E
=
∇
E
+
θ,
where
θ
is
a
square
differential.
Let
ι
:
M
→
E
and
ι
:
M
→
E
be
the
respective
inclusions,
and
χ
:
M
→
N
and
χ
:
M
→
N
the
respective
composites
discussed
in
the
preceding
paragraph.
We
claim
first
of
all
that
M
and
M
are
isomorphic.
Indeed,
this
follows
from
the
fact
that
N
⊗
M
−1
∼
=
N
⊗
(M
)
−1
.
=
O
X
(E
E
)
=
O
X
(E
E
)
∼
Thus,
we
shall
henceforth
identify
M
and
M
.
Now
χ
and
χ
differ
by
multiplication
by
a
section
λ
of
O
S
,
that
is,
χ
=
λ
·
χ
.
Let
i
1
:
M
→
E
be
ι,
and
let
i
2
:
M
→
E
be
ι
multiplied
by
λ.
Then
it
follows
that
there
exists
a
morphism
α
:
M
→
F
1
(E)
such
that
i
1
=
i
2
+
α.
Now
let
s
be
a
horizontal
section
of
M
(over
some
open
set
U
⊆
X).
Since
the
p-curvature
is
a
horizontal
morphism,
∇
E
(i
1
(s))
=
0
and
∇
E
(i
2
(s))
=
0.
Thus,
we
compute:
∇
E
(i
2
(s))
=
(∇
E
+
θ)(i
1
(s)
+
α(s))
=
∇
E
(α(s))
+
θ(i
1
(s))
Suppose
that
α
is
nonzero.
Then
in
the
last
line,
the
first
term
has
a
nonzero
image
under
the
surjection
E
→
N
(since
the
Kodaira-Spencer
morphism
is
an
isomorphism),
while
the
second
term
lies
in
F
1
(E).
Since
the
sum
of
these
terms
is
zero,
we
thus
obtain
a
contradiction.
Thus,
α
must
be
zero.
Then
we
obtain
that
θ(i
1
(s))
=
0,
so
(if
θ
=
0,
then)
i
1
maps
into
F
1
(E)
⊆
E.
But
then
χ
=
0,
so
by
(2),
we
again
obtain
a
contradiction.
This
completes
the
proof
of
(4).
Proposition
2.7.
Suppose
that
S
is
reduced.
Let
(E,
∇
E
)
be
admissible
indigenous
on
X
log
.
Let
P
E
:
T
→
Ad(E)
be
the
p-curvature.
Let
U
⊆
X
be
open,
and
let
(L,
∇
L
)
be
a
line
bundle
with
logarithmic
connection
(relative
to
f
log
)
on
U
whose
p-curvature
is
zero.
Let
ι
:
L
→
Ad(E)
be
a
horizontal
morphism.
Then
ι
factors
through
T
∼
=
Im(P
E
).
Proof.
By
shrinking
U
,
we
may
assume
that
L
and
T
|
U
have
horizontal
generating
sections
s
and
t,
respectively.
We
may
assume
that
s
and
t
generate
a
subbundle
G
⊆
Ad(E)|
U
of
rank
two.
If
we
take
their
commutator
in
Ad(E),
we
see
that
[s,
t]
must
be
in
G.
Indeed,
if
this
were
not
the
case,
then
the
p-curvature
of
Ad(E)
would
be
zero.
But
the
p-curvature
of
Ad(E)
is
given
by
Ad(P
E
)
which
is
nonzero
everywhere
since
(E,
∇
E
)
is
admissible.
This
proves
the
claim.
Thus,
G
is
a
Lie
subalgebra
of
Ad(E),
stabilized
by
the
connection
on
Ad(E)
and
whose
p-curvature
is
zero.
Being
of
rank
two,
it
is
necessarily
solvable,
hence
contains
a
nilpotent
subalgebra
K
⊆
G
which
is
stabilized
by
the
connection
on
Ad(E)
and
has
p-curvature
zero.
Now
K
defines
a
horizontal
filtration
N
⊆
E|
U
with
respect
to
which
it
is
nilpotent.
Since
(N
)
⊗2
∼
=
K,
it
follows
that
N
has
p-curvature
zero.
Let
δ
:
End(E|
U
)
→
Ad(E)|
U
be
the
canonical
projection
given
by
quotienting
out
by
the
scalar
endomorphisms.
Then
clearly
the
image
of
N
⊗
E|
U
⊆
End(E|
U
)
(i.e.,
the
endomorphisms
that
anihilate
N
⊆
E|
U
)
under
δ
is
equal
to
G.
Since
G
and
N
have
p-curvature
zero,
it
thus
follows
that
E|
U
has
p-curvature
zero,
which
contradicts
the
fact
that
(E,
∇
E
)
is
admissible.
Proposition
2.8.
Suppose
that
f
log
is
obtained
by
gluing
together
various
f
i
log
,
as
in
70
the
last
subsection
of
Chapter
I,
§2.
Suppose
that
(E,
∇
E
)
is
nilpotent,
admissible,
and
indigenous
on
X
log
.
Then
it
is
automatically
of
restrictable
type.
Proof.
The
subbundle
M
⊆
E
(of
Proposition
2.5)
is
stabilized
by
E
and
has
p-curvature
zero.
Thus,
if
we
restrict
to
an
irreducible
component
X
i
log
,
the
monodromy
at
any
marked
point
of
X
i
log
must
be
nilpotent
with
respect
to
the
filtration
defined
by
M
⊆
E.
This
completes
the
proof.
As
mentioned
earlier,
the
reason
that
we
are
interested
in
nilpotent,
admissible
indige-
nous
bundles
is
that
they
define
MF
∇
-objects
in
the
sense
of
[Falt].
Let
us
suppose
that
S
=
Spec(k),
where
k
is
a
perfect
field,
and
that
f
:
X
→
S
is
smooth.
Let
S
=
Spec(A),
where
A
=
W
(k)/p
2
W
(k),
and
W
(k)
is
the
ring
of
Witt
vectors
with
coefficients
in
k.
Let
We
suppose
that
S
and
S
are
us
denote
by
Φ
A
the
canonical
Frobenius
morphism
on
S.
endowed
with
the
trivial
log
structures,
and
call
the
resulting
log
schemes
S
log
and
S
log
,
log
→
S
log
be
a
smooth
r-pointed
curve
of
genus
g
that
lifts
f
log
.
respectively.
Let
f
log
:
X
⊆
X
be
Let
(E,
∇
E
)
be
an
indigenous
bundle
on
X
log
.
Let
E
=
F
1
(E)
⊕
(E/F
1
(E)).
Let
U
log
→
U
log
be
a
lifting
of
Frobenius.
If
e
is
a
section
of
E,
an
open
subset,
and
let
Φ
log
:
U
Φ
∗
1
let
∇
E
(e)
be
the
section
of
Φ
(E/F
(E))
⊗
ω
U
log
/S
log
obtained
by
applying
∇
E
to
e
(and
regarding
the
result
modulo
F
1
(E)
⊗
ω
U
log
/S
log
)
to
get
a
section
of
(E/F
1
(E))
⊗
ω
U
log
/S
log
,
then
pulling
back
by
Φ
on
E
and
by
p
1
dΦ
on
ω
U
log
/S
log
.
Then
we
can
define
a
logarithmic
connection
∇
Φ
on
Φ
∗
(
E
F
)
by
letting
∇
Φ
(Φ
−1
(0,
e
F
))
=
0
(if
e
is
a
section
of
E/F
1
(E))
1
and
∇
Φ
(Φ
−1
(e
F
,
0))
=
(0,
∇
Φ
E
(e))
(if
e
is
a
section
of
F
(E)).
Then,
it
is
easy
to
see
(as
in
[Falt],
§2)
that
for
different
Φ,
the
pairs
(Φ
∗
(
E
F
)|
U
,
∇
Φ
)
glue
together
to
form
a
bundle
with
connection
F
∗
(E,
∇
E
)
on
X
log
.
Note
that
F
∗
(E,
∇
E
)
depends
on
the
choice
of
lifting
log
→
S
log
.
X
Definition
2.9.
We
shall
say
that
(E,
∇
E
)
forms
an
MF
∇
-object
on
X
log
if
(E,
∇
E
)
⊗
(L,
∇
L
)
∼
=
F
∗
(E,
∇
E
)
log
→
S
log
and
some
line
bundle
with
connection
(L,
∇
L
)
whose
for
some
choice
of
lifting
X
square
is
trivial.
Note
that
this
definition
is
consistent
(though
slightly
weaker,
since
we
allow
the
ambiguity
of
tensoring
with
(L,
∇
L
))
with
the
notion
of
being
an
object
of
the
category
MF
∇
of
[Falt],
§2.
It
is
shown
in
[Falt]
(Theorem
6.2)
that
the
de
Rham
cohomology
of
a
semistable
family
of
varieties
over
X
log
equipped
with
the
Hodge
filtration
and
the
Gauss-Manin
connection
forms
an
object
of
MF
∇
(as
long
as
p−
2
is
greater
than
or
equal
to
the
relative
dimension
of
the
family
of
varieties).
The
following
result
provides
the
link
between
what
we
are
doing
here
and
[Falt],
§2:
71
Proposition
2.10.
Let
(E,
∇
E
)
be
an
indigenous
bundle
on
X
log
.
Then
(E,
∇
E
)
forms
an
MF
∇
-object
on
X
log
if
and
only
if
(E,
∇
E
)
is
admissible
and
nilpotent.
Proof.
First,
let
us
assume
that
(E,
∇
E
)
forms
an
MF
∇
-object
on
X
log
for
the
lifting
log
→
S
log
.
One
computes
easily
from
the
definition
of
the
connection
∇
Φ
that
the
p-
X
Also,
(just
curvature
is
nilpotent
(with
respect
to
the
filtration
0
⊕
(Φ
∗
E/F
1
(E))
⊆
Φ
∗
E).
1
as
in
the
proof
of
Proposition
1.4),
the
derivative
p
Φ
is
of
the
form
(1
+
t)
p−1
+
f
(t).
Thus,
since
the
Kodaira-Spencer
morphism
is
the
identity,
it
follows
that
the
p-curvature
def
(applied
to
∂
=
d/dt)
is
obtained
by
multiplying
a
section
of
(Φ
∗
F
1
(E))
⊕
0
by
Φ
−1
(∂
F
)
times
the
(p
−
1)
th
derivative
of
(1
+
t)
p−1
+
f
(t)
(which
is
just
−1)
and
regarding
the
result
of
this
multiplication
as
a
section
of
0
⊕
(Φ
∗
E/F
1
(E)).
In
particular,
we
see
that
the
p-curvature
is
nonzero.
Thus,
(E,
∇
E
)
is
admissible
and
nilpotent.
On
the
other
hand,
suppose
that
(E,
∇
E
)
is
admissible
and
nilpotent.
By
the
bijective
correspondence
of
Proposition
2.5,
(E,
∇
E
)
corresponds
to
an
FL-bundle
(G,
∇
G
).
By
log
→
S
log
.
It
remains
to
see
Proposition
1.2,
this
FL-bundle
corresponds
to
a
lifting
X
∗
that
F
(E,
∇
E
)
taken
with
respect
to
this
lifting
is
isomorphic
to
(E,
∇
E
)
(up
to
tensoring
with
an
(L,
∇
L
)
whose
square
is
trivial).
Let
N
=
E/F
1
(E).
Let
M
=
Φ
∗
X/S
(N
F
).
Let
∇
M
be
the
connection
on
M
for
which
the
sections
of
N
F
are
horizontal.
Thus,
the
fact
that
the
(M,
∇
M
)
⊗2
∼
=
(T
,
∇
T
).
Now,
sorting
through
the
definitions
and
using
Kodaira-Spencer
morphism
is
the
identity
reveals
that
if
X
=
U
V
(where
U
and
V
are
affine
opens),
then
F
∗
(E,
∇
E
)
⊗
(M,
∇
M
)
is
just
the
extension
of
O
X
by
T
obtained
from
the
1-cocycle
which
is
the
difference
between
Frobenius
liftings
on
U
and
V
.
It
thus
follows
from
the
definition
of
the
canonical
morphism
F
:
D
→
A
of
Proposition
1.2
that
F
∗
(E,
∇
E
)
⊗
(M,
∇
M
)
is
exactly
the
bundle
(G,
∇
G
).
Thus,
it
follows
from
the
definition
of
the
bijective
correspondence
in
Proposition
2.5
that
F
∗
(E,
∇
E
)
∼
=
(E,
∇
E
)
⊗
(L,
∇
L
)
for
some
(L,
∇
L
)
whose
square
is
trivial.
The
Infinitesimal
Verschiebung
Let
ω
X/S
be
the
relative
dualizing
sheaf
of
f
:
X
→
S.
Thus,
if
D
⊆
X
is
the
divisor
of
log
marked
points,
we
have
ω
X/S
(D)
∼
=
ω
X/S
.
Let
Φ
X/S
:
X
→
X
F
be
the
relative
Frobenius
over
S.
Recall
from
duality
theory
(see,
e.g.,
[Harts]
for
a
treatment
of
duality
theory)
that
since
f
and
Φ
X/S
are
local
complete
intersection
morphisms,
we
have
a
trace
morphism:
tr
Φ
X/S
:
(Φ
X/S
)
∗
ω
X/S
→
(ω
X/S
)
F
where
we
regard
ω
X/S
as
RΦ
!
X/S
(ω
X/S
)
F
.
On
the
other
hand,
let
Y
log
be
the
log
scheme
whose
underlying
scheme
is
X
and
whose
log
structure
is
the
same
as
that
of
X
log
away
from
the
divisor
D
of
marked
points,
and
equal
to
the
pull-back
of
the
log
structure
of
S
log
on
the
open
subscheme
where
f
is
smooth.
Then
we
also
have
a
morphism
arising
from
the
log
version
of
the
Cartier
isomorphism
(applied
to
Y
log
→
S
log
):
72
C
:
(Φ
X/S
)
∗
ω
X/S
→
(ω
X/S
)
F
The
following
result
is
“well-known,”
but
I
do
not
know
an
adequate
reference
for
it:
Lemma
2.11.
The
morphisms
tr
Φ
X/S
and
C
are
equal.
Proof.
By
a
density
argument,
we
reduce
to
the
case
where
f
is
smooth.
By
naturality,
we
reduce
to
the
assertion
that
these
two
morphisms
are
the
same
when
X
=
Spec(F
p
[T
]);
S
=
Spec(F
p
)
(where
T
is
an
indeterminate).
Since
tr
Φ
X/S
is
the
reduction
modulo
p
of
a
construction
that
holds
in
arbitrary
or
mixed
characteristic,
we
can
calculate
tr
Φ
X/S
by
considering
the
trace
map
obtained
from
duality
for
the
finite
morphism
Φ
:
Spec(Z
p
[T
])
→
Spec(Z
p
[T
])
given
by
T
→
T
p
.
Since
tr
Φ
(Φ
∗
(dT
))
=
p
dT
,
and
Φ
∗
(dT
)
=
p
T
p−1
dT
,
it
follows
that
tr
Φ
(T
p−1
dT
)
=
dT
.
By
reducing
this
formula
modulo
p
and
comparing
with
the
construction
in
[Katz]
of
the
Cartier
isomorphism,
we
obtain
the
desired
result.
Let
(E,
∇
E
)
be
an
indigenous
bundle
on
X
log
.
Let
H
E
:
T
→
τ
X
log
/S
log
be
the
square
Hasse
invariant
of
(E,
∇
E
).
Then
by
pulling
back
via
Φ
X
log
/S
log
and
taking
R
1
f
∗
of
H
E
,
we
obtain
a
canonical
morphism:
Φ
τ
E
:
(R
1
f
∗
τ
X
log
/S
log
)
F
→
R
1
f
∗
τ
X
log
/S
log
which
we
shall
call
the
Frobenius
on
R
1
f
∗
τ
X
log
/S
log
induced
by
(E,
∇
E
).
On
the
other
log
→
T
∨
.
Note
that
hand,
let
us
consider
the
dual
morphism
to
H
E
,
that
is,
H
E
∨
:
ω
X/S
log
F
T
∨
=
Φ
∗
X/S
(ω
X/S
)
.
Thus,
if
we
tensor
H
E
∨
with
ω
X/S
,
we
get
a
morphism
log
⊗2
log
F
)
(−D)
→
ω
X/S
⊗
O
X
Φ
∗
X/S
(ω
X/S
)
(ω
X/S
log
F
If
we
then
compose
this
morphism
with
the
trace
morphism
tr
Φ
X/S
tensored
with
(ω
X/S
)
,
log
⊗2
log
⊗2
we
obtain
a
morphism
(Φ
X/S
)
∗
(ω
X/S
)
(−D)
→
((ω
X/S
)
(−D))
F
.
Then
applying
f
∗
to
this
morphism,
we
obtain:
log
⊗2
log
⊗2
(−D)
→
(f
∗
(ω
X/S
)
(−D))
F
Φ
ω
E
:
f
∗
(ω
X/S
)
log
⊗2
which
we
shall
call
the
Verschiebung
on
f
∗
(ω
X/S
)
(−D)
induced
by
(E,
∇
E
).
Observe
that
by
Serre
duality
applied
to
the
morphism
f
,
we
obtain
that
the
vector
bundles
log
⊗2
)
(−D)
and
R
1
f
∗
τ
X
log
/S
log
on
S
are
dual
to
one
another.
Then
relative
to
this
f
∗
(ω
X/S
duality,
we
have
τ
Proposition
2.12.
The
morphisms
Φ
ω
E
and
Φ
E
are
dual
to
one
another.
73
Proof.
This
follows
immediately
from
duality
theory.
Namely,
the
duality
between
⊗2
(−D)
and
R
1
f
∗
τ
X
log
/S
log
is
obtained
by
cup
product,
followed
by
the
trace
mor-
f
∗
ω
X/S
phism
tr
f
:
R
1
f
∗
ω
X/S
→
O
S
.
On
the
other
hand,
since
trace
morphisms
behave
well
under
composition,
we
see
that
tr
f
=
(tr
f
)
F
◦
tr
Φ
X/S
.
This
fact,
combined
with
the
fact
∨
that
Φ
τ
E
(respectively,
Φ
ω
E
)
is
obtained
from
H
E
(respectively,
H
E
),
implies
the
result.
We
are
now
ready
to
state
and
prove
the
second
main
result
of
this
Section.
Consider
the
Verschiebung
on
indigenous
bundles
defined
in
the
first
subsection
of
this
Section.
In
the
universal
case,
it
was
a
morphism
V
g,r
:
S
g,r
→
Q
g,r
over
M
g,r
.
Thus,
it
induces
a
map
on
tangent
bundles
over
M
g,r
:
Θ
V
g,r
/M
g,r
:
Θ
S
g,r
/M
g,r
→
Θ
Q
g,r
/M
g,r
If
we
pull-back
to
the
point
of
S
g,r
defined
by
our
particular
(E,
∇
E
),
we
get
a
morphism
log
⊗2
log
⊗2
Θ
V
(−D)
→
(f
∗
(ω
X/S
)
(−D))
F
E
:
f
∗
(ω
X/S
)
Then
we
have
the
following
result:
Theorem
2.13.
Let
(E,
∇
E
)
be
indigenous
on
X
log
.
Then
the
infinitesimal
Verschiebung
ω
morphism
Θ
V
E
is
equal
to
Φ
E
.
Proof.
First,
let
us
recall
Jacobson’s
formula
(see,
e.g.,
[Jac],
pp.
186-187):
This
formula
states
that
if
a
and
b
are
elements
of
an
associative
ring
R
of
characteristic
p,
then
p
p
p
(a
+
b)
=
a
+
b
+
p−1
s
i
(a,
b)
i=1
where
the
s
i
(a,
b)
are
given
by
the
formula:
p−1
(ad(ta
+
b))
(a)
=
p−1
is
i
(a,
b)t
i−1
i=1
computed
in
the
ring
R[t],
where
t
is
an
indeterminate.
In
our
case,
we
wish
to
apply
this
formula
in
the
case
where
a
=
α,
α
∈
R,
and
is
an
element
of
the
center
of
R
such
that
2
=
0.
In
this
case,
substitution
yields:
74
(a
+
b)
p
=
b
p
+
(ad(b))
p−1
a
ω
To
prove
that
Θ
V
E
=
Φ
E
,
it
suffices
to
do
a
local
calculation
on
X
to
show
that
the
infinitesimal
change
in
the
trace
of
the
square
of
the
p-curvature
is
given
by
−2
Φ
ω
E
.
We
log
⊗2
2
work
over
the
base
S[
]/(
).
Let
∇
E
=
∇
E
+
θ,
where
θ
is
a
section
of
f
∗
(ω
X/S
)
(−D),
be
a
connection
on
E
that
makes
it
an
indigenous
bundle.
Let
U
⊆
X
be
an
open
subset
that
avoids
the
marked
points
and
at
which
f
is
smooth.
Let
x
be
a
local
coordinate
on
U
.
def
d
.
Write
∇
(respectively,
∇
;
θ
x
)
for
∇
E
(respectively,
∇
E
;
θ)
applied
in
the
direction
∂
=
dx
We
wish
to
apply
the
above
formula
in
the
case
where
b
=
∇
and
a
=
θ
x
.
We
thus
obtain
that
the
infinitesimal
change
in
the
p-curvature
is
given
by
(ad(∇))
p−1
θ
x
.
Now
the
infinitesimal
Verschiebung
is
obtained
by
multiplying
this
term
by
the
constant
term
and
then
taking
the
trace
(and
multiplying
by
−1).
Put
another
way,
minus
the
infinitesimal
Verschiebung
is
obtained
by
applying
P
E
∨
to
(ad(∇))
p−1
θ
x
.
Since
P
E
∨
is
horizontal,
it
commutes
with
ad(∇),
so
we
find
that:
(∂)
p−1
(P
E
∨
θ
x
)
=
(∂)
p−1
((P
E
∨
θ)
·
∂)
log
F
)
obtained
by
evaluating
minus
the
infinitesimal
Verschiebung
is
the
section
of
Φ
∗
X/S
(ω
X/S
log
⊗2
(which
is
a
section
of
Φ
∗
X/S
{((ω
X/S
)
)(−D)}
F
)
at
∂
F
.
Thus,
to
complete
the
proof
of
the
Theorem,
we
must
show
that
∨
F
(∂)
p−1
((P
E
∨
θ)
·
∂)
=
−Φ
−1
X/S
{(C
⊗
1)(P
E
θ)
·
∂
}
log
F
)
.
Write
P
E
∨
θ
=
ν
·
ζ,
Now
let
ζ
be
a
horizontal,
locally
generating
section
of
Φ
∗
X/S
(ω
X/S
where
ν
is
a
section
of
ω
X/S
.
Substituting,
we
see
that
it
suffices
to
show
that:
F
(∂)
p−1
(ν
·
∂)
·
ζ
=
−Φ
−1
X/S
(C(ν)
·
∂
)
·
ζ
If
we
then
divide
out
the
ζ’s,
we
see
that
really
(just
as
in
[Katz2],
(7.1.2.6)),
we
are
reduced
to
proving
a
simple
identity
concerning
differentiation
in
characteristic
p.
Indeed,
if
we
regard
the
equation
as
an
identity
in
ν,
both
sides
are
Φ
−1
X/S
O
X
F
-linear
in
ν
and
vanish
when
ν
is
exact;
thus,
we
are
reduced
to
proving
the
identity:
(∂)
p−1
(x
p−1
)
=
−C(x
p−1
dx)
·
∂
F
which
follows
from
the
definition
of
C
and
the
fact
that
(p
−
1)!
=
−1
in
characteristic
p.
75
Differential
Criterion
for
Admissibility
We
maintain
the
notation
of
the
previous
subsection.
In
the
previous
subsection,
we
computed
the
relative
differential
map
of
the
Verschiebung
morphism
V
g,r
over
M
g,r
.
In
fact,
however,
with
a
little
more
preparation,
the
same
calculation
allows
us
to
give
an
explicit
representation
of
the
differential
map
of
the
Verschiebung
over
F
p
.
Moreover,
this
explicit
representation
allows
us
to
give
a
differential
criterion
for
an
indigenous
bundle
to
be
admissible,
which
is
also
necessary
if
the
bundle
is
nilpotent.
Consider
the
affine
bundle
Q
g,r
→
M
g,r
.
Since
by
definition,
this
bundle
is
the
pull
back
by
the
Frobenius
morphism
on
M
g,r
of
a
bundle
over
M
g,r
,
it
follows
that
we
get
log
a
natural
connection
∇
Q
on
this
affine
bundle
Q
g,r
→
M
g,r
.
Let
Q
g,r
be
the
log
stack
log
obtained
by
pulling
back
to
Q
g,r
the
log
structure
of
M
g,r
.
Next,
let
us
consider
the
canonical
exact
sequence
of
tangent
bundles
on
Q
g,r
:
0
→
Θ
Q
g,r
/M
g,r
→
Θ
Q
log
→
Θ
M
log
|
Q
g,r
→
0
g,r
g,r
Thus,
our
connection
∇
Q
induces
a
splitting
Θ
Q
log
∼
=
Θ
Q
g,r
/M
g,r
⊕
Θ
M
log
|
Q
g,r
g,r
g,r
Now
let
us
consider
the
“full”
infinitesimal
Verschiebung,
i.e.,
the
morphism
induced
on
tangent
bundles
by
V
g,r
:
Θ
V
g,r
:
Θ
S
g,r
→
Θ
Q
g,r
On
the
one
hand,
we
know
what
the
projection
of
Θ
V
g,r
to
Θ
M
log
|
Q
g,r
is.
Thus,
we
would
g,r
like
to
compute
the
projection
of
Θ
V
g,r
to
Θ
Q
g,r
/M
g,r
.
We
shall
soon
see
that,
in
fact,
we
have
already
computed
this
projection
as
well,
in
the
course
of
proving
Theorem
2.13.
Suppose
(as
in
the
previous
subsection)
we
have
a
log
scheme
S
log
with
an
r-pointed
stable
curve
of
genus
g,
f
log
:
X
log
→
S
log
,
and
an
indigenous
bundle
(E,
∇
E
)
on
X
log
.
Then
the
first
relative
parabolic
de
Rham
cohomology
module
R
1
f
DR,∗
(Ad(E))
(as
in
Chapter
I,
Theorem
2.8)
is
naturally
isomorphic
to
the
pull-back
to
S
log
(via
the
classifying
morphism
for
{X
log
,
(E,
∇
E
)})
of
Θ
S
log
.
On
R
1
f
DR,∗
(Ad(E)),
we
have
a
Hodge
filtration
g,r
log
⊗2
0
→
f
∗
(ω
X/S
)
(−D)
→
R
1
f
DR,∗
(Ad(E))
→
R
1
f
∗
τ
X
log
/S
log
→
0
where
the
surjection
in
the
above
exact
sequence
is
exactly
the
pull-back
to
S
log
of
the
projection
Θ
S
log
→
Θ
M
log
|
S
g,r
.
g,r
g,r
76
On
the
other
hand,
consider
the
p-curvature
of
Ad(E):
P
E
:
T
→
Ad(E).
Then
by
applying
“R
1
f
DR,∗
”
to
the
dual
of
P
E
,
we
get
a
morphism
R
1
f
DR,∗
(Ad(E))
→
R
1
f
DR,∗
(T
∨
)
Now
by
Poincaré
duality,
we
have
R
1
f
DR,∗
(T
∨
)
∼
=
{R
1
f
DR,∗
(T
)}
∨
Moreover,
we
computed
R
1
f
DR,∗
(T
)
in
Proposition
1.1.
In
particular,
we
have
a
natural
inclusion
R
1
f
∗
(τ
X
log
/S
log
)
F
→
R
1
f
DR,∗
(T
).
Thus,
if
we
compose
the
above
morphism
induced
by
P
E
∨
with
the
dual
surjection
to
this
natural
inclusion,
we
obtain
a
natural
morphism
log
⊗2
)
(−D))
F
Θ
E
:
R
1
f
DR,∗
(Ad(E))
→
(f
∗
(ω
X/S
Then
we
have
the
following
result:
Theorem
2.14.
Let
(E,
∇
E
)
be
indigenous
on
X
log
.
Then
the
pull-back
to
S
log
of
the
projection
of
Θ
V
g,r
to
Θ
M
log
is
given
by
the
surjection
in
the
Hodge
filtration;
the
pull-
g,r
back
to
S
log
of
the
projection
of
Θ
V
g,r
to
Θ
Q
g,r
/M
g,r
is
given
by
Θ
E
.
Proof.
It
remains
to
prove
the
statement
about
the
projection
to
Θ
Q
g,r
/M
g,r
.
To
do
this
we
consider
an
infinitesimal
deformation
of
(E,
∇
E
)
over
S
log
[
]/(
2
).
But
the
section
of
log
⊗2
)
(−D))
F
that
we
obtain
can
be
computed
locally
on
X.
Moreover,
locally
on
(f
∗
(ω
X/S
X,
this
calculation
is
exactly
the
same
as
that
of
Theorem
2.13.
This
proves
the
result.
Corollary
2.15.
Let
(E,
∇
E
)
be
indigenous
on
X
log
.
If
Θ
E
is
surjective,
then
(E,
∇
E
)
is
log
log
log
admissible.
In
particular,
if
the
morphism
V
g,r
:
S
g,r
→
Q
g,r
is
log
étale
at
the
image
of
log
the
classifying
morphism
S
log
→
S
g,r
for
(E,
∇
E
),
then
(E,
∇
E
)
is
admissible.
Proof.
Since
being
admissible
is
an
open
condition,
it
suffices
to
prove
the
result
when
S
log
is
the
spectrum
of
an
algebraically
closed
field.
If
V
g,r
is
log
étale,
then
by
Theorem
2.14,
Θ
E
must
be
surjective.
Thus,
it
suffices
to
prove
the
first
statement.
Suppose
that
(E,
∇
E
)
is
not
admissible.
Thus,
the
morphism
P
E
∨
:
Ad(E)
→
T
∨
has
a
nonempty
zero
locus.
One
can
compute
the
p-curvature
explicitly
at
a
marked
point
(where
the
monodromy
is
nonzero
and
nilpotent);
it
follows
that
the
zero
locus
does
not
contain
any
marked
points.
Now
there
are
two
possibilities:
the
zero
locus
either
avoids
the
nodes
or
it
does
not.
Let
us
first
do
the
case
where
the
zero
locus
of
P
E
∨
avoids
the
nodes.
Since
P
E
∨
is
horizontal,
its
zero
locus
must
be
the
pull-back
of
some
closed
subscheme
via
Φ
X/S
.
Thus,
77
in
particular,
there
exists
some
point
x
∈
X
(which
is
neither
a
marked
point
nor
a
F
node)
such
that
P
E
∨
is
zero
at
Φ
−1
X/S
(x
)
(the
scheme-theoretic
fiber).
By
the
definition
of
log
⊗2
)
(−D))
F
Θ
E
,
it
follows
that
the
image
of
Θ
E
lands
in
the
subspace
V
x
of
H
0
(X,
(ω
X/S
consisting
of
sections
that
vanish
at
x
F
.
Now
by
Riemann-Roch
on
curves,
V
x
cannot
be
log
⊗2
all
of
H
0
(X,
(ω
X/S
)
(−D))
F
,
unless
g
=
0
and
r
=
3,
or
g
=
1
and
r
=
1.
This
completes
the
proof
(under
the
assumption
that
P
E
∨
avoids
the
nodes),
except
for
these
two
special
cases.
For
g
=
0,
r
=
3,
we
shall
show
in
the
subsection
of
§3
on
totally
degenerate
curves
that
the
unique
indigenous
bundle
on
such
a
curve
is
necessarily
admissible.
(One
checks
easily
that
there
are
no
vicious
circles
in
the
reasoning.)
For
g
=
1,
r
=
1,
we
note
that
P(E,
∇
E
)
is
necessarily
invariant
with
respect
to
the
automorphism
α
given
by
multiplying
by
−1.
Thus,
if
we
pull-back
by
the
morphism
X
→
X
given
by
multiplying
by
2,
it
is
still
invariant
under
α.
Hence
it
descends
to
the
four-pointed
curve
of
genus
zero
Y
log
of
which
X
log
is
a
log
étale
double
covering.
Let
us
call
this
descended
bundle
P(F,
∇
F
).
Thus,
P(F,
∇
F
)
is
indigenous
on
Y
log
.
It
is
easy
to
see
that
Θ
E
and
Θ
F
are
the
same
morphism;
thus,
the
hypothesis
holds
for
P(F,
∇
F
),
as
well.
Thus,
we
reduce
to
the
case
g
=
0,
r
=
4,
which
has
already
been
checked.
Now
let
us
consider
the
case
where
P
E
∨
vanishes
at
a
node
ν
∈
X.
Let
π
log
:
Z
log
→
X
log
be
the
partial
normalization
of
X
log
at
ν
(where
the
log
structure
on
Z
log
is
such
def
that
the
points
mapping
to
ν
are
marked
points).
Let
(F,
∇
F
)
=
π
∗
(E,
∇
E
).
Now
let
us
consider
the
commutative
diagram:
H
1
(X,
τ
X
log
/S
log
)
F
⏐
⏐
−→
1
H
DR
(X
log
,
Ad(E))
⏐
⏐
−→
H
1
(X,
τ
X
log
/S
log
)
⏐
⏐
H
1
(Z,
τ
Z
log
/S
log
)
F
−→
1
H
DR
(Z
log
,
Ad(F))
−→
H
1
(Z,
τ
Z
log
/S
log
)
where
the
vertical
arrows
are
pull-backs
via
π;
the
horizontal
arrows
on
the
left
are
induced
by
P
E
;
and
the
horizontal
arrows
on
the
right
are
induced
by
the
Hodge
filtration.
Finally,
the
“prime”
on
the
de
Rham
cohomology
on
the
bottom
row
indicates
that
the
we
are
taking
non-parabolic
de
Rham
cohomology
on
Z
log
(since
(F,
∇
F
)
may
not
even
have
a
natural
parabolic
structure,
if
(E,
∇
E
)
is
not
of
restrictable
type).
Let
η
be
a
generator
of
the
kernel
of
H
1
(X,
τ
X
log
/S
log
)
→
H
1
(Z,
τ
Z
log
/S
log
).
Let
us
consider
what
happens
to
η
F
as
we
move
it
around
the
above
commutative
diagram.
Since
P
E
is
zero
at
ν,
if
we
move
it
1
(X
log
,
Ad(E))
lies
in
along
the
top
to
H
1
(X,
τ
X
log
/S
log
),
we
get
zero.
Thus,
its
image
in
H
DR
log
⊗2
1
1
)
(−D)).
But
the
pull-back
map
H
DR
(X
log
,
Ad(E))
→
H
DR
(Z
log
,
Ad(F))
H
0
(X,
(ω
X/S
log
⊗2
log
⊗2
maps
H
0
(X,
(ω
X/S
)
(−D))
injectively
into
H
0
(Z,
(ω
Z/S
)
).
Going
around
the
other
way,
however,
(i.e.,
going
down
and
then
to
the
right),
we
see
that
the
image
of
η
F
in
1
1
H
DR
(Z
log
,
Ad(F))
is
zero.
We
thus
conclude
that
the
image
of
η
F
in
H
DR
(X
log
,
Ad(E))
is
zero.
But
this
contradicts
the
surjectivity
of
Θ
E
,
since
the
upper
horizontal
morphism
on
the
left-hand
side
of
the
above
diagram
is
dual
to
Θ
E
.
This
completes
the
proof.
78
Conversely,
let
us
suppose
that
(E,
∇
E
)
is
nilpotent
and
admissible.
Then
we
claim
that
Θ
E
is
necessarily
surjective.
Indeed,
for
simplicity,
it
suffices
to
prove
this
when
S
is
the
spectrum
of
a
field.
Let
K
be
the
kernel
of
P
E
∨
:
Ad(E)
→
T
∨
.
Note
that
the
connection
on
Ad(E)
restricts
to
a
connection
∇
K
on
K.
Thus,
(K,
∇
K
)
is
an
FL-bundle.
Also,
we
have
a
horizontal
exact
sequence:
0
→
T
→
K
→
O
X
→
0
in
which
the
connecting
morphism
O
S
→
R
1
f
DR,∗
(T
)
must
be
injective
(since
(K,
∇
K
)
is
an
FL-bundle).
Moreover,
it
is
a
tautology
that
the
composite
of
this
morphism
O
S
→
R
1
f
DR,∗
(T
)
with
the
projection
R
1
f
DR,∗
(T
)
→
O
S
of
Proposition
1.1
is
the
identity.
Thus,
we
see
that
the
morphism
R
1
f
∗
(τ
X
log
/S
log
)
F
→
R
1
f
DR,∗
(T
)
→
R
1
f
DR,∗
(K)
is
injective.
Now
let
C
=
Ad(E)/T
.
We
have
a
connection
∇
C
on
C,
induced
by
the
connection
on
Ad(E).
Thus,
we
get
a
horizontal
exact
sequence:
0
→
O
X
→
C
→
T
∨
→
0
log
F
)
→
R
1
f
DR,∗
(O
X
)
must
be
injective
(since
in
which
the
connecting
morphism
(f
∗
ω
X/S
otherwise,
C
would
admit
two
horizontal,
generically
linearly
indendent
sections,
which
contradicts
the
fact
that
the
p-curvature
is
nonzero).
Lastly,
we
consider
the
horizontal
exact
sequence:
0
→
K
→
Ad(E)
→
T
∨
→
0
log
F
)
→
R
1
f
DR,∗
(K)
with
the
in
which
the
composite
of
the
connecting
morphism
(f
∗
ω
X/S
projection
R
1
f
DR,∗
(K)
→
R
1
f
DR,∗
(O
X
)
is
injective,
as
we
observed
above.
Since
the
image
of
R
1
f
∗
(τ
X
log
/S
log
)
F
⊆
R
1
f
DR,∗
(K)
under
the
map
R
1
f
DR,∗
(K)
→
R
1
f
DR,∗
(O
X
)
is
zero,
it
thus
follows
that
if
we
compose
the
inclusion
R
1
f
∗
(τ
X
log
/S
log
)
F
⊆
R
1
f
DR,∗
(K)
with
the
morphism
R
1
f
DR,∗
(K)
→
R
1
f
DR,∗
(Ad(E)),
the
resulting
morphism
R
1
f
∗
(τ
X
log
/S
log
)
F
→
R
1
f
DR,∗
(Ad(E))
is
injective.
But
this
morphism
is
dual
to
Θ
E
.
Thus,
Θ
E
is
surjective.
This
completes
the
proof
of
the
claim.
log
log
Let
N
g,r
be
the
log
stack
obtained
by
pulling
back
the
log
structure
on
M
g,r
to
N
g,r
.
log
Since
N
g,r
is
the
zero
locus
of
the
Verschiebung,
it
follows
that
N
g,r
is
log
smooth
over
F
p
at
a
point
if
and
only
if
Θ
E
is
surjective.
In
other
words:
Corollary
2.16.
Suppose
that
(E,
∇
E
)
is
nilpotent
indigenous.
Then
it
is
admissible
if
log
and
only
if
N
g,r
is
smooth
over
F
p
at
the
image
of
classifying
morphism
S
log
→
N
g,r
for
(E,
∇
E
).
79
§3.
Hyperbolically
Ordinary
Curves
Often,
in
the
literature,
one
speaks
of
a
curve
as
being
“ordinary”
if
its
Jacobian
is
ordinary.
In
fact,
however,
since
the
Jacobian
only
represents
the
“abelian
part”
of
the
curve,
it
is
in
some
sense
more
intrinsic
to
speak
of
a
curve
as
ordinary
if
it
is
hyperbolically
ordinary
in
the
sense
defined
below.
Philosophically,
this
means
that
the
Verschiebung
on
indigenous
bundles
is
a
local
isomorphism
in
a
neighborhood
of
an
indigenous
bundle
that
provides
a
“nice”
uniformization
for
the
curve.
Thus,
relative
to
the
analogy
(explained
in
the
Introduction)
between
the
Verschiebung
on
indigenous
bundles
and
the
Beltrami
equa-
tion,
to
be
hyperbolically
ordinary
means
that
the
Verschiebung
acts
(at
least
locally)
as
one
might
expect
from
this
analogy,
given
the
classical
results
on
existence
and
uniqueness
of
solutions
to
the
Beltrami
equation.
Basic
Definitions
Let
S
log
be
a
fine
noetherian
log
scheme
over
F
p
.
Let
f
log
:
X
log
→
S
log
be
an
r-pointed
stable
curve
of
genus
g
(so
2g
−
2
+
r
≥
1).
Let
D
⊆
X
be
the
divisor
of
marked
points.
Let
(E,
∇
E
)
be
an
indigenous
bundle
on
X
log
.
Let
P
E
:
T
→
Ad(E)
be
its
p-curvature.
Recall
the
Frobenius
on
R
1
f
∗
τ
X
log
/S
log
induced
by
(E,
∇
E
):
Φ
τ
E
:
(R
1
f
∗
τ
X
log
/S
log
)
F
→
R
1
f
∗
τ
X
log
/S
log
that
we
defined
in
§2.
Definition
3.1.
We
shall
call
(E,
∇
E
)
ordinary
if
Φ
τ
E
is
an
isomorphism.
Note
that
the
condition
of
being
ordinary
is
an
open
condition
on
S
g,r
.
We
shall
denote
ord
this
open
set
by
S
g,r
.
Proposition
3.2.
If
(E,
∇
E
)
is
ordinary,
then
it
is
admissible.
Proof.
This
follows
a
fortiori
from
Corollary
2.15.
The
following
definition
is
key
to
the
entire
paper:
Definition
3.3.
We
shall
say
that
f
log
:
X
log
→
S
log
is
a
hyperbolically
ordinary
curve
if
there
exists
an
étale
surjection
T
→
S
and
a
nilpotent,
ordinary
indigenous
bundle
(E,
∇
E
)
on
X
log
×
S
T
.
80
When
the
context
is
clear,
we
shall
simply
say
that
f
log
is
an
“ordinary
curve.”
The
reason
for
the
descriptive
“hyperbolically”
is
that
in
the
literature,
the
term
“ordinary
curve”
is
frequently
used
to
mean
that
its
Jacobian
is
ordinary.
In
this
paper,
when
f
log
has
an
ordinary
Jacobian,
we
shall
say
that
f
log
is
parabolically
ordinary.
Proposition
3.4.
If
the
fiber
of
f
log
:
X
log
→
S
log
over
s
∈
S
is
hyperbolically
ordinary,
then
there
exists
an
open
set
U
⊆
S
with
s
∈
U
such
that
f
log
|
U
is
hyperbolically
ordinary.
In
particular,
f
log
:
X
log
→
S
log
is
hyperbolically
ordinary
if
and
only
if
all
its
fibers
are
hyperbolically
ordinary.
Proof.
Indeed,
it
suffices
to
consider
the
universal
example.
Recall
that
N
g,r
⊆
S
g,r
is
the
locus
of
nilpotent
indigenous
bundles.
Write
π
:
N
g,r
→
M
g,r
for
the
natural
projection.
Let
n
∈
N
g,r
;
let
m
∈
M
g,r
be
the
point
π(n).
Then
it
follows
from
Theorem
2.13,
plus
the
definition
of
N
g,r
as
the
zero
locus
of
V
g,r
that
if
n
is
ordinary,
then
π
must
be
étale
at
n.
Thus,
π
is
open
at
n.
This
completes
the
proof.
We
shall
denote
the
open
subscheme
of
M
g,r
(respectively,
N
g,r
)
consisting
of
hyper-
ord
bolically
ordinary
curves
(respectively,
nilpotent,
ordinary
indigenous
bundles)
by
M
g,r
ord
(respectively,
N
g,r
).
Thus,
we
have
a
natural
étale
surjection
ord
ord
π
:
N
g,r
→
M
g,r
ord
Finally,
let
us
note
that
over
S
g,r
,
we
have
an
étale
local
system
in
F
p
-vector
spaces
of
dimension
3g
−
3
+
r
obtained
by
taking
the
sections
of
Θ
M
log
|
S
ord
that
are
invariant
g,r
g,r
under
the
Frobenius
action
on
Θ
M
log
|
S
ord
given
by
−Φ
τ
E
.
(Note
the
minus
sign
in
front
of
g,r
g,r
−Φ
τ
E
!
It
will
be
important
later
in
Chapter
III.)
Let
us
denote
this
local
system
by
Θ
et
g,r
,
ord
and
call
it
the
tangential
local
system
on
S
g,r
.
Similarly,
by
taking
its
dual
Ω
et
g,r
we
obtain
ord
ord
a
local
system
on
S
g,r
which
we
shall
call
the
differential
local
system
on
S
g,r
.
Often,
we
shall
be
interested
in
the
restrictions
of
these
local
systems
to
N
g,r
.
The
Totally
Degenerate
Case
In
this
subsection,
we
show
that
totally
degenerate
curves
are
hyperbolically
ordinary.
ord
By
Proposition
3.4,
this
will
show
that
M
g,r
is
an
open
dense
subscheme
of
M
g,r
.
Since
totally
degenerate
curves
have
no
moduli,
there
is
no
loss
of
generality
in
assuming
that
S
=
Spec(F
p
).
We
begin
by
considering
the
case
g
=
0,
r
=
3.
Recall
the
morphisms
constructed
at
the
end
of
Chapter
I,
§3:
log
N
log
:
M
1,1
[2]
→
M
1,1
81
(parametrizing
elliptic
curves
with
level
structures
on
the
two-torsion
points)
and
log
log
Λ
log
:
M
1,1
[2]
→
M
0,4
which
takes
an
elliptic
curve
to
the
four-pointed
curve
of
genus
zero
of
which
it
is
a
log
double
covering.
Then
in
this
case,
we
have
M
0,4
=
X
log
.
Let
us
construct
a
nilpotent,
log
admissible
indigenous
bundle
on
M
0,4
.
Since
X
log
has
only
one
indigenous
P
1
-bundle
(up
to
isomorphism),
this
will
complete
the
proof
of
Corollary
2.15.
To
do
this,
we
note
that
(as
we
saw
in
Example
3
of
Chapter
I,
§2),
the
first
de
Rham
cohomology
module
of
log
the
universal
elliptic
curve
over
M
1,1
[2]
defines
an
indigenous
vector
bundle
(E,
∇
E
).
Let
(P
→
X,
∇
P
)
be
the
associated
P
1
-bundle.
Now
since
the
map
“multiplication
by
−1”
on
an
elliptic
curve
induces
the
map
“multiplication
by
−1”
on
E,
it
follows
that
(E,
∇
E
)
will
not
descend
via
Λ.
However,
since
“multiplication
by
−1”
induces
the
identity
on
P
→
X,
we
see
that
(P
→
X,
∇
P
)
does
descend
via
Λ.
This
gives
us
an
indigenous
bundle
on
X
log
.
To
see
that
it
is
nilpotent
and
admissible,
it
suffices
to
see
that
(E,
∇
E
)
is
nilpotent
and
admissible.
But
by
(a
rather
trivial
special
case
of)
[Falt],
Theorem
6.2,
as
a
de
Rham
cohomology
module,
(E,
∇
E
)
necessarily
forms
an
MF
∇
-object
(see
Definition
2.9).
Thus,
Proposition
2.10
tells
us
that
(E,
∇
E
)
is
nilpotent
and
admissible.
In
particular,
this
completes
the
proof
of
Corollary
2.15.
Now
let
us
assume
that
f
log
:
X
log
→
S
log
is
formed
by
gluing
together
a
number
of
copies
of
the
3-pointed
stable
curve
of
genus
zero
(as
in
the
last
subsection
of
Chapter
I,
§2).
Then,
as
we
saw
in
this
final
subsection
of
Chapter
I,
§2,
we
can
glue
together
the
nilpotent,
admissible
indigenous
bundles
that
we
constructed
in
the
previous
paragraph
to
obtain
a
nilpotent,
admissible
indigenous
bundle
(P
→
X,
∇
P
)
on
X
log
.
On
the
other
hand,
by
Proposition
2.8,
every
nilpotent,
admissible
indigenous
bundle
on
X
log
is
of
restrictable
type.
Since
there
is
(up
to
isomorphism)
only
one
indigenous
P
1
-bundle
of
restrictable
type
on
X
log
,
it
thus
follows
that:
Proposition
3.5.
Up
to
isomorphism,
a
totally
degenerate
r-pointed
stable
curve
of
genus
g
admits
one
and
only
one
nilpotent,
admissible
indigenous
P
1
-bundle.
Next,
let
us
consider
the
cohomology
group
H
1
(X,
τ
X
log
/S
log
)
of
our
totally
degenerate
curve.
If
X
log
is
obtained
by
gluing
together
various
copies
X
i
log
of
the
3-pointed
stable
curve
of
genus
zero,
let
Y
log
be
the
disjoint
union
of
the
X
i
log
,
and
let
ν
log
:
Y
log
→
X
log
be
the
natural
map.
Let
0
→
τ
X
log
/S
log
→
ν
∗
τ
Y
log
/S
log
→
C
→
0
be
the
natural
exact
sequence
(where
C
is
defined
so
as
to
make
the
sequence
exact).
By
considering
the
long
exact
cohomology
sequence
associated
to
this
exact
sequence
of
sheaves,
we
see
that
we
obtain
a
natural
isomorphism
H
1
(X,
τ
X
log
/S
log
)
∼
=
H
0
(X,
C).
On
the
other
hand,
C
is
naturally
isomorphic
to
the
direct
sum
of
the
(τ
X
log
/S
log
)|
z
,
where
z
ranges
over
all
the
nodes
in
X.
Moreover,
the
residue
map
gives
a
natural
isomorphism
τ
X
log
/S
log
|
z
∼
=
O
S
(well-defined
up
to
sign).
Let
Σ
be
the
set
of
nodes
of
X.
Then
82
C
=
(O
S
)
z
z∈Σ
where
the
subscript
“z”
is
just
used
as
a
marker,
to
indicate
which
copy
of
O
S
one
is
referring
to.
Thus,
we
have
a
natural
isomorphism
(well-defined
up
to
sign
on
each
factor):
H
1
(X,
τ
X
log
/S
log
)
∼
=
(F
p
)
z
z∈Σ
In
particular,
since
F
p
has
a
natural
bilinear
form
(given
by
ring
multiplication
F
p
×
F
p
→
F
p
),
using
this
bilinear
form
on
each
factor
(F
p
)
z
gives
a
natural
bilinear
form:
B
:
H
1
(X,
τ
X
log
/S
log
)
×
H
1
(X,
τ
X
log
/S
log
)
→
F
p
which
is
now
independent
of
all
arbitrary
choices
of
sign.
Proposition
3.6.
For
every
totally
degenerate
r-pointed
curve
of
genus
g,
there
is
a
natural
nondegenerate
bilinear
form
B
on
the
O
S
-module
R
1
f
∗
τ
X
log
/S
log
which
takes
values
in
O
S
.
Next
we
would
like
to
show
that
the
unique
nilpotent,
admissible
indigenous
bundle
(P
→
X,
∇
P
)
on
X
log
is
ordinary.
To
do
this,
we
must
compute
the
induced
Frobenius
action
on
H
1
(X,
τ
X
log
/S
log
).
By
using
an
isomorphism
as
above
H
1
(X,
τ
X
log
/S
log
)
∼
=
(F
p
)
z
z∈Σ
we
see
that
it
suffices
to
compute
the
induced
Frobenius
action
on
the
various
(F
p
)
z
’s.
Consider
P
|
z
.
The
Hodge
section
σ
:
X
→
P
defines
a
point
σ
z
∈
P
|
z
(F
p
).
On
the
other
hand,
there
is
a
unique
point
fixed
by
the
monodromy
action
q
z
∈
P
|
z
(F
p
).
If
we
think
of
P
as
P(J
/J
[3]
)
(as
in
Chapter
I,
Proposition
2.5),
then
(J
/J
[3]
)
z
=
V
σ
⊕
V
q
,
where
V
σ
(respectively,
V
q
)
is
the
subspace
defined
by
σ
z
(respectively,
q
z
).
Note
that
by
the
residue
map,
we
have
natural
isomorphisms
V
σ
∼
=
F
p
and
V
q
∼
=
F
p
.
Thus,
we
obtain
a
basis
[3]
{(1,
0);
(0,
1)}
of
(J
/J
)
z
.
Let
E
σ
and
E
q
be
the
nilpotent
endomorphisms
of
(J
/J
[3]
)
z
given,
respectively,
by
the
matrices
0
1
0
0
1
0
and
0
0
Thus,
E
q
is
essentially
the
p-curvature
of
(π
:
P
→
X,
∇
P
)
restricted
to
z.
Sorting
through
all
the
definitions,
it
thus
follows
that
the
induced
Frobenius
action
on
(F
p
)
z
is
given
by
multiplication
by
tr(E
σ
·
E
q
)
=
1.
We
thus
obtain
the
following
result:
83
Proposition
3.7.
On
a
totally
degenerate
r-pointed
stable
curve
X
log
of
genus
g
over
F
p
,
the
Frobenius
action
Φ
τP
on
H
1
(X,
τ
X
log
/S
log
)
induced
by
the
unique
nilpotent,
admissible
indigenous
bundle
on
X
log
is
the
identity.
In
particular,
this
unique
nilpotent,
admissible
indigenous
bundle
is
ordinary,
and
so
is
X
log
.
ord
ord
Corollary
3.8.
The
open
subschemes
M
g,r
⊆
M
g,r
and
N
g,r
⊆
N
g,r
are
nonempty.
The
Case
of
Elliptic
Curves:
The
Parabolic
Picture
One
can
get
a
better
feel
for
ordinariness
for
general
r-pointed
stable
curves
of
genus
g
by
first
studying
ordinariness
for
elliptic
curves.
In
the
case
of
elliptic
curves,
there
are,
in
fact,
two
possible
theories
of
ordinary
bundles
and
curves:
the
parabolic
theory
and
the
hyperbolic
theory.
Indeed,
let
f
log
:
X
log
→
S
log
be
a
1-pointed
stable
curve
of
genus
1,
with
marked
point
:
S
→
X.
Let
Y
log
be
the
log
scheme
obtained
from
X
log
(as
in
the
subsection
“The
Infinitesimal
Verschiebung”
of
§2)
by
removing
the
marked
point.
Then
the
parabolic
theory
(respectively,
hyperbolic
theory)
is
obtained
by
considering
the
various
properties
of
the
p-curvature
of
indigenous
bundles
on
Y
log
(respectively,
X
log
).
So
far
in
this
Chapter,
of
course,
we
have
only
been
considering
the
hyperbolic
theory.
However,
since
the
notion
of
an
indigenous
bundle
is
defined
for
Y
log
,
one
can
consider
its
p-curvature,
and
define
the
notions
of
a
nilpotent
indigenous
bundle,
or
an
admissible
indigenous
bundle,
just
as
before.
Also,
many
of
the
results
(though
not
all)
such
as
Theorem
2.3
(where
we
replace
the
“3g
−
3
+
r”
by
1)
continue
to
hold
in
the
parabolic
context.
The
purpose
of
this
subsection
is
to
summarize
what
happens
when
one
studies
elliptic
curves
from
the
parabolic
point
of
view,
and
to
show,
in
particular,
that
the
notion
of
ordinariness
that
we
have
defined
in
this
paper
(in
terms
of
the
p-curvature
of
indigenous
bundles)
reduces
to
the
classical
notion
of
ordinariness
of
elliptic
curves.
log
log
First,
we
introduce
some
notation.
Let
M
1,0
=
M
1,1
.
(The
point
here
is
that
we
log
shall
use
the
notation
M
1,0
when
we
are
thinking
about
elliptic
curves
from
the
parabolic
point
of
view.)
Let
f
:
G
→
M
1,0
be
the
universal
elliptic
curve,
with
identity
section
:
M
1,0
→
G.
Let
L
=
f
∗
ω
G/M
1,0
be
the
Hodge
bundle.
Let
G
log
be
the
log
stack
whose
log
underlying
stack
is
G
and
whose
log
structure
is
the
pull-back
of
the
log
structure
on
M
1,0
.
Let
S
1,0
→
M
1,0
be
the
torsor
over
L
⊗2
of
Schwarz
structures
on
G
log
.
Then
just
as
before,
we
can
define
a
Verschiebung:
V
1,0
:
S
1,0
→
Q
1,0
Just
as
before
we
have
a
closed
subscheme
N
1,0
⊆
S
1,0
consisting
of
nilpotent
indigenous
adm
bundles,
and
an
open
subscheme
S
1,0
⊆
S
1,0
consisting
of
admissible
indigenous
bundles.
Also,
just
as
before,
we
define
an
indigenous
bundle
to
be
ordinary
if
its
infinitesimal
Ver-
schiebung
is
an
isomorphism,
and
we
define
an
elliptic
curve
to
be
(parabolically)
ordinary
if
it
admits
a
nilpotent,
ordinary
indigenous
bundle.
84
Now
recall
that
in
Example
2
of
Chapter
I,
§2,
we
constructed
a
canonical
indigenous
bundle
on
G
log
.
This
indigenous
bundle
thus
defines
a
global
section
τ
S
:
M
1,0
→
S
1,0
which
trivializes
the
L
⊗2
-torsor
S
1,0
→
M
1,0
.
Moreover,
by
[KM],
p.
227,
one
knows
that
if
p
≥
5,
then
H
0
(M
1,0
,
L
⊗2
)
=
0,
so
this
trivialization
is
unique.
Let
us
also
recall
that,
in
the
definition
of
the
indigenous
bundle
(E,
∇
E
)
in
Example
2
of
Chapter
I,
§2,
we
had
a
subbundle
0
⊕
O
G
⊆
ω
⊕
O
G
=
E
(where
ω
=
ω
G/M
1,0
)
which
was
stabilized
by
the
connection
∇
E
.
Moreover,
the
induced
connection
on
O
G
was
the
trivial
connection.
Put
another
way,
E
admits
a
nonzero
horizontal
section.
It
thus
follows
that
the
p-curvature
of
∇
E
is
nilpotent.
Thus,
τ
S
:
M
1,0
→
S
1,0
lands
inside
N
1,0
.
Let
θ
be
a
section
of
L
⊗2
over
some
étale
V
→
M
1,0
.
Let
∇
θ
E
be
the
connection
formed
by
adding
to
∇
E
the
endomorphism
given
by
E
=
ω
⊕
O
G
→
O
G
→
ω
⊗2
∼
=
(ω
⊕
0)
⊗
ω
⊆
E
⊗
ω
where
the
first
morphism
is
the
natural
projection;
the
second
morphism
is
multiplication
by
θ;
and
the
final
inclusion
is
the
natural
one.
Let
δ
be
a
section
of
L
over
V
which
is
everywhere
nonzero.
Thus,
the
sections
δ
and
1
define
a
global
trivialization
of
E
=
ω
⊕
O
G
over
G
V
.
We
shall
write
sections
of
E|
V
in
terms
of
this
basis,
given
by
δ
and
1.
Write
θ
=
φ
·
δ
2
,
where
φ
is
a
function
on
V
.
Let
∇
θ
be
the
morphism
E|
V
→
E|
V
given
by
evaluating
∇
θ
E
on
δ
−1
.
Then
we
see
that
∇
θ
is
given
by
the
following
matrix:
0
φ
1
0
To
obtain
the
p-curvature
of
∇
θ
E
,
we
must
iterate
this
matrix
p
times.
This
yields
the
matrix:
φ
1
2
(p−1)
·
0
φ
1
0
Now
let
us
write
δ
−p
=
h
·
δ
−1
(so
h
is
the
classical
Hasse
invariant).
Thus,
to
compute
the
p-curvature
of
∇
θ
E
,
we
must
subtract
from
the
matrix
just
given
the
following
matrix:
h
·
0
φ
1
0
Thus,
we
obtain
that
the
p-curvature
of
∇
θ
E
is
given
by:
(φ
1
2
(p−1)
−
h)
·
85
0
φ
1
0
If
we
then
take
the
determinant,
we
obtain
our
Verschiebung
(applied
to
(δ
F
)
2
):
1
1
−φ
·
(φ
2
(p−1)
−
h)
2
=
−φ
p
+
2h
·
φ
2
(p+1)
−
h
2
·
φ
Let
us
rewrite
this
in
invariant
form.
The
trivialization
τ
S
of
S
1,0
→
M
1,0
allows
us
to
write
S
1,0
as
Spec(⊕
i≥0
L
⊗−2i
).
On
the
other
hand,
Q
1,0
is
given
by
Spec(⊕
i≥0
L
⊗−2p·i
).
Thus,
V
1,0
is
determined
by
specifying
the
morphism
of
quasi-coherent
sheaves:
Γ
V
:
L
⊗−2p
→
L
⊗−2i
i≥0
Let
us
denote
the
component
of
Γ
V
that
maps
into
L
⊗−2i
by
Γ
V
:
L
⊗−2p
→
L
⊗−2i
.
Since
[i]
L
is
ample,
it
follows
that
Γ
V
=
0
when
i
>
p
(as
we
saw
already
in
the
proof
of
Theorem
2.3).
Let
χ
∈
Γ(M
1,0
,
L
p−1
)
be
the
Hasse
invariant
(as
in
[KM],
p.
353).
Then
we
see
that
we
have
proven
the
following
result:
[i]
[i]
[i]
Theorem
3.9.
If
i
=
p,
then
Γ
V
is
multiplication
by
−1.
If
i
=
12
(p
+
1),
then
Γ
V
[i]
is
multiplication
by
2
χ.
If
i
=
1,
then
Γ
V
is
multiplication
by
−χ
2
.
For
all
other
i,
[i]
Γ
V
=
0.
In
particular,
this
completes
the
proof
of
Theorem
2.3.
Corollary
3.10.
Geometrically,
N
1,0
consists
of
two
irreducible
components
I
1
and
I
2
:
One,
I
1
,
is
the
section
τ
S
.
The
other,
I
2
,
is
nonreduced,
and
(I
2
)
red
may
be
described
as
follows:
In
[KM],
p.
361,
one
finds
a
description
of
the
Igusa
curve
Ig(p),
with
its
canonical
(Z/pZ)
×
-action.
Then
(I
2
)
red
is
the
quotient
of
Ig(p)
by
the
subgroup
{±1}
⊆
(Z/pZ)
×
.
In
order
to
see
which
nilpotent
bundles
are
ordinary,
we
must
compute
the
derivative
of
the
Verschiebung
map.
In
terms
of
the
local
objects
we
used
in
the
computation
above,
we
obtain
that
“dV/dφ”
is
given
by:
1
h(φ
2
(p−1)
−
h)
In
particular,
if
h
=
0,
then
the
infinitesimal
Verschiebung
is
identically
zero,
while
if
h
=
0,
then
the
infinitesimal
Verschiebung
at
τ
S
is
nonzero.
Moreover,
because
of
the
square
factor
in
the
expression
for
the
Verschiebung,
we
see
that
if
h
=
0,
then
the
only
nilpotent
indigenous
bundle
at
which
the
infinitesimal
Verschiebung
will
be
nonzero
is
the
indigenous
bundle
given
by
τ
S
.
We
thus
obtain
the
following:
Theorem
3.11.
An
elliptic
curve
is
parabolically
ordinary
if
and
only
if
it
is
ordinary
in
the
classical
sense
(i.e.,
its
Hasse
invariant
is
nonzero).
If
it
is
ordinary,
then
the
indigenous
bundle
constructed
in
Example
2
of
Chapter
I,
§2,
is
the
unique
nilpotent,
ordinary
indigenous
bundle
on
the
curve.
86
The
Case
of
Elliptic
Curves:
The
Hyperbolic
Picture
In
this
subsection,
we
consider
1-pointed
stable
curves
of
genus
1
as
hyperbolic
objects.
In
particular,
we
shall
highlight
the
numerous
contrasts
with
the
parabolic
viewpoint
presented
above.
We
begin
by
considering
the
torsor
S
1,1
→
M
1,1
.
Recall
that
this
torsor
has
a
canonical
trivialization
at
infinity,
defined
by
the
unique
nilpotent,
admissible
indigenous
bundle
on
the
singular
1-pointed
stable
curve
of
genus
1
(Proposition
3.5).
Let
us
suppose
that
our
prime
p
is
≥
5.
Then
it
follows
from
Chapter
I,
Theorem
3.6,
that
S
1,1
→
M
1,1
does
not
admit
a
section
which
passes
through
the
canonical
trivialization
at
infinity.
Now
let
us
consider
the
closed
subscheme
N
1,1
⊆
S
1,1
.
By
Theorem
2.3,
the
natural
morphism
N
1,1
→
M
1,1
is
finite
and
flat
of
degree
p.
Let
us
consider
the
irreducible
component
I
⊆
N
1,1
which
passes
through
the
canonical
trivialization
at
infinity.
Then
I
is
generically
reduced.
Moreover,
the
degree
of
I
→
M
1,1
must
be
≥
2.
This
behavior
already
is
substantially
different
from
the
parabolic
case,
where
the
irreducible
component
passing
through
the
unique
nilpotent,
ordinary
indigenous
bundle
at
infinity
has
degree
one
over
M
1,0
.
Thus,
in
particular,
N
1,1
(respectively,
V
1,1
)
is
not
isomorphic
to
N
1,0
(respectively,
V
1,0
),
despite
the
fact
that
as
stacks,
M
1,1
∼
=
M
1,0
;
S
1,1
∼
=
S
1,0
;
Q
1,1
∼
=
Q
1,0
.
Since
N
1,1
→
M
1,1
has
degree
p,
it
follows
that
there
exist
points
of
N
1,1
over
the
infinity
point
of
M
1,1
at
which
N
1,1
→
M
1,1
is
not
étale.
Such
points
correspond
to
nilpotent
indigenous
bundles
which
are
not
admissible
(by
Proposition
3.5).
This
fulfills
our
earlier
pledge
to
show
the
existence
of
such
bundles.
Of
these
various
observations,
we
record
the
following
for
later
reference:
Proposition
3.12.
If
p
≥
5,
then
the
irreducible
component
of
N
1,1
passing
through
the
canonical
trivialization
at
infinity
is
generically
reduced
and
has
degree
≥
2
over
M
1,1
.
The
Generic
Uniformization
Number
We
return
to
the
case
of
an
r-pointed
stable
curve
of
genus
g,
where
r
and
g
are
arbitrary
(but
satisfy
2g
−
2
+
r
≥
1).
Suppose
we
are
given
the
combinatorial
data
Δ
(as
at
the
end
of
Chapter
I,
Secion
2:
consisting
of
a
graph
Γ,
plus
λ
i
’s,
etc.)
for
a
totally
degenerate
curve.
We
shall
call
two
collections
of
such
data
Δ
and
Δ
equivalent
if
they
define
isomorphic
totally
degenerate
curves.
Let
us
denote
by
D
g,r
the
equivalence
classes
of
such
data
Δ.
Alternatively,
one
may
think
of
D
g,r
as
the
set
of
isomorphism
classes
of
totally
degenerate
r-pointed
stable
curves
of
genus
g.
Now
let
us
consider
the
morphism:
N
g,r
→
M
g,r
We
know
that
it
is
finite,
flat,
and
of
degree
p
3g−3+r
.
If
Δ
∈
D
g,r
,
consider
the
irreducible
component
I
Δ
of
N
g,r
that
passes
through
the
unique
nilpotent,
admissible
indigenous
bundle
(as
in
Proposition
3.5)
on
the
curve
corresponding
to
Δ.
Then
I
Δ
is
generically
reduced.
Let
G
Δ
be
the
degree
of
I
Δ
over
M
g,r
.
We
shall
refer
to
G
Δ
as
the
generic
87
uniformization
number
for
the
data
Δ.
(The
reason
for
attaching
the
term
“uniformization”
to
this
number
will
become
apparent
in
later
Chapters.)
Let
G
g,r
=
G
Δ
Δ∈D
g,r
Now
let
us
suppose
that
p
is
sufficiently
large
so
that
the
class
Σ
(Chapter
I,
Theorem
3.4)
in
H
1
(M
g,r
,
Ω
M
log
)
is
nonzero.
Then
we
have
the
following
rough
result:
g,r
Proposition
3.13.
For
g
≥
3
and
p
sufficiently
large,
the
number
G
Δ
is
between
2
and
p
3g−3+r
.
Proof.
The
upper
bound
follows
from
the
fact
that
N
g,r
→
M
g,r
has
degree
p
3g−3+r
.
On
the
other
hand,
since
N
g,r
→
M
g,r
is
finite,
and
M
g,r
is
normal,
it
follows
that
if
I
Δ
had
degree
1
over
M
g,r
,
it
would,
in
fact,
be
isomorphic
to
M
g,r
,
hence
define
a
section
of
S
g,r
over
M
g,r
.
By
Chapter
I,
Theorem
3.4,
we
know
that
this
is
impossible,
for
p
sufficiently
large.
It
is
not
clear
to
the
author
how
far
these
bounds
are
from
being
sharp.
For
instance,
it
could
be
the
case
that
N
g,r
is,
in
fact,
irreducible.
To
compute
the
number
G
Δ
exactly
would
involve
understanding
the
monodromy
around
curves
that
are
not
hyperbolically
ordinary.
That
is
to
say,
it
would
involve
proving
a
sort
of
hyperbolic
analogue
of
Igusa’s
theorem
on
the
monodromy
around
supersingular
elliptic
curves
in
the
parabolic
case.
It
is
interesting
to
know,
however,
that
G
Δ
=
1
because
this
constitutes
a
depar-
ture
from
the
behavior
of
complex
indigenous
bundles.
To
see
this,
we
must
first
explain
certain
aspects
of
the
analogy
between
the
complex
case
and
the
characteristic
p
case
treated
here.
First
of
all,
the
condition
of
being
nilpotent
(and
ordinary)
is
analogous,
in
the
complex
case,
to
having
real
monodromy.
Indeed,
to
be
nilpotent
(and
admissible)
is
(by
Proposition
2.10)
the
same
as
coming
from
an
MF
∇
-object
in
the
sense
of
[Falt].
But
to
be
an
MF
∇
-object
means,
essentially,
that
the
bundle
with
connection
admits
a
Frobenius
action,
i.e.,
that
the
monodromy
is
Frobenius-invariant.
Since
the
Frobenius
at
the
infinite
prime
is
complex
conjugation,
it
is
thus
natural
to
regard
nilpotent
(and
admissible)
bundles
as
the
characteristic
p
analogue
of
complex
indigenous
bundles
with
real
monodromy.
On
the
other
hand,
in
the
complex
case,
within
the
real-analytic
space
of
complex
indigenous
bundles
with
real
monodromy,
there
is
a
canonical,
topologically
isolated
component,
corresponding
to
the
indigenous
bundle
arising
from
the
uniformiza-
tion
by
the
upper
half-plane.
On
the
other
hand,
in
the
characteristic
p
case
treated
here,
the
fact
that
I
Δ
has
degree
≥
2
over
M
g,r
means
that
there
is
no
canonical
choice
of
a
nilpotent,
ordinary
indigenous
bundle,
even
on
a
generic
curve:
since
the
monodromy
at
the
curves
which
are
not
hyperbolically
ordinary
is
nontrivial,
one
such
indigenous
bundle
is
always
carried
around
to
another.
88
Chapter
III:
Canonical
Modular
Frobenius
Liftings
§0.
Introduction
The
present
Chapter
is
central
to
the
entire
paper.
In
it,
we
construct
a
canonical
ord
Frobenius
lifting
on
(N
g,r
)
log
,
and
a
canonical
indigenous
bundle
on
the
universal
curve
ord
over
(N
g,r
)
log
.
This
pair
of
a
canonical
Frobenius
lifting
and
a
canonical
indigenous
bundle
are
uniquely
characterized
by
the
fact
that,
relative
to
this
Frobenius
lifting,
the
renormalized
Frobenius
pull-back
(Definition
1.4)
of
the
canonical
indigenous
bundle
is
equal
to
itself.
In
some
sense,
an
ordinary
(in
the
sense
of
Definition
1.1)
Frobenius
lifting
is
like
a
p-adic
analogue
of
a
Kähler
metric
on
a
complex
manifold
in
that
it
gives
rise
to
local
canonical
coordinates.
Since
there
are
a
number
of
general
properties
of
ordinary
Frobenius
liftings
that
we
will
need
throughout
the
rest
of
the
paper,
we
give
a
basic
exposition
of
the
properties
of
such
Frobenius
liftings
in
the
first
Section
of
this
Chapter.
The
main
result
is
that
such
a
Frobenius
lifting
defines
canonical
affine
and
multiplicative
ord
coordinates.
Thus,
in
particular,
our
canonical
Frobenius
lifting
on
(N
g,r
)
log
defines
such
ord
ord
canonical
coordinates
on
N
g,r
.
Since
N
g,r
is
étale
over
M
g,r
,
if
one
thinks
of
a
point
ord
of
N
g,r
as
a
point
of
M
g,r
,
together
with
a
choice
(from
a
finite
number
of
possibilities)
of
some
added
structure
–
which
we
call
a
p-adic
quasiconformal
equivalence
class
–
then
we
obtain
the
result
that
for
every
choice
of
a
p-adic
quasiconformal
equivalence
class,
we
obtain
a
canonical
local
uniformization
of
M
g,r
.
The
reason
for
the
name
“quasiconformal
equivalence
class,”
is
that
once
one
chooses
this
piece
of
data
for
a
curve,
we
shall
see
in
this
Chapter
and
in
following
Chapters
that
the
uniformization
theory
of
the
curve
is
entirely
determined.
This
is
reminiscent
of
Bers’
approach
(as
in
[Bers])
to
proving
that
hyperbolic
curves
can
be
uniformized
by
the
upper
half
plane:
Namely,
he
proves
that
(in
the
complex
case)
all
hyperbolic
curves
with
the
same
genus
and
number
of
marked
points
belong
to
the
same
quasiconformal
equivalence
class.
Thus,
once
we
choose
this
class
in
the
p-adic
case,
we
obtain
a
“covariant”
uniformization
by
the
affine
space
modeled
on
the
tangent
space
to
M
g,r
at
the
curve
in
question.
To
obtain
uniformizations
by
the
quadratic
differentials
(as
in
the
complex
case),
we
need
more
information
than
just
the
quasiconformal
equivalence
class.
Namely,
we
need
a
topological
marking
of
the
curve.
Once
we
define
this,
we
obtain
uniformizations
by
the
quadratic
differentials.
The
canonical
Frobenius
gives
rise
to
another
natural
notion,
for
which
I
know
no
parallel
in
the
complex
case:
a
canonical
lifting
of
a
curve
over
a
perfect
field
to
the
ring
of
Witt
vectors
with
coefficients
in
that
field.
This
is
reminiscent
of
the
canonical
lifting
of
an
elliptic
curve
in
Serre-Tate
theory.
In
fact,
this
analogy
is
more
than
philosophical:
Just
as
in
Chapter
II,
by
using
indigenous
bundles
on
elliptic
curves
–
regarded
parabolically
–
one
can
obtain
a
similar
uniformization
theory,
involving
a
canonical
Frobenius
lifting
(on
the
moduli
stack
of
ordinary
elliptic
curves)
and
a
canonical
indigenous
bundle.
We
then
compute
that
these
canonical
objects
for
elliptic
curves
are
precisely
the
canonical
objects
89
that
one
obtains
from
classical
Serre-Tate
theory.
Thus,
one
may
regard
the
theory
of
uniformizations
and
canonical
liftings
discussed
in
this
Chapter
as
the
natural
hyperbolic
analogue
of
Serre-Tate
theory.
§1.
Generalities
on
Ordinary
Frobenius
Liftings
Let
k
be
a
perfect
field
of
odd
characteristic
p.
Let
A
=
W
(k),
the
ring
of
Witt
vectors
with
coefficients
in
k.
Let
S
be
a
formally
smooth,
geometrically
connected
p-adic
formal
scheme
over
A
of
constant
relative
dimension
d.
Let
S
log
be
a
log
formal
scheme
whose
underlying
formal
scheme
is
S
and
whose
log
structure
is
given
by
a
relative
divisor
with
normal
crossings
D
⊆
S
over
A.
Let
Φ
A
:
A
→
A
be
the
Frobenius
morphism
on
A.
Let
us
denote
the
result
of
base
changing
by
Φ
A
by
means
of
a
superscripted
“F
.”
Let
Φ
log
:
S
log
→
S
log
be
a
Frobenius
lifting,
i.e.,
a
morphism
whose
reduction
modulo
p
is
the
usual
Frobenius
morphism
in
characteristic
p.
In
this
Section,
we
shall
study
the
case
of
a
certain
kind
of
Frobenius
lifting,
called
an
ordinary
Frobenius
lifting.
It
turns
out
that
such
Frobenius
liftings
define
various
types
of
canonical
parameters.
It
is
these
canonical
parameters
that
will
constitute
the
various
“uniformizations”
that
we
obtain
in
this
paper.
Basic
Definitions
Let
us
consider
the
morphism
log
dΦ
log
:
Φ
∗
Ω
log
S/A
→
Ω
S/A
induced
by
Φ
log
on
logarithmic
differentials.
It
is
always
divisible
by
p.
Definition
1.1.
We
shall
call
Φ
log
:
S
log
→
S
log
an
ordinary
Frobenius
lifting
if
p
1
·
dΦ
log
:
log
Φ
∗
Ω
log
S/A
→
Ω
S/A
is
an
isomorphism.
We
shall
use
the
notation
log
Ω
Φ
:
Φ
∗
Ω
log
S/A
→
Ω
S/A
to
denote
the
isomorphism
p
1
·
dΦ
log
.
Note
that
by
considering
the
sections
of
Ω
log
S/A
which
are
invariant
under
Ω
Φ
,
we
obtain
an
étale
sheaf
Ω
et
Φ
in
free
Z
p
-modules
of
rank
d
on
S.
Definition
1.2.
We
shall
call
Ω
et
Φ
the
canonical
differential
local
system
on
S
associated
to
Φ
log
.
We
shall
call
its
dual
Θ
et
Φ
the
canonical
tangential
local
system
on
S
associated
to
log
Φ
.
90
Moreover,
by
taking
the
sections
of
Ω
et
Φ
to
be
horizontal,
we
obtain
a
natural
connection
log
log
∇
Ω
on
Ω
S/A
which
is
associated
to
Φ
.
Note
that
since
Ω
et
Φ
is
an
étale
(not
just
log
étale)
local
system
on
S,
the
connection
∇
Ω
is
a
connection
on
S
with
respect
to
the
trivial
log
structure,
i.e.,
it
has
no
logarithmic
poles
at
D.
The
Uniformizing
Galois
Representation
Now
we
would
like
to
associate
to
Φ
log
a
canonical
“uniformizing
MF
∇
-object”
(where
we
use
the
category
MF
∇
in
the
sense
of
[Falt],
§2)
as
follows.
Let
P
=
Ω
log
S/A
⊕
O
S
We
regard
P
as
being
filtered
by
taking
the
filtration
def
def
def
0
F
2
(P)
=
0;
F
1
(P)
=
Ω
log
S/A
⊕
0
⊆
P;
F
(P)
=
P
Let
∇
P
be
the
logarithmic
connection
on
P
obtained
as
follows:
We
start
with
the
connec-
tion
∇
P
on
P
which
is
the
direct
sum
of
the
trivial
connection
on
O
S
and
the
connection
∇
Ω
on
Ω
log
S/A
.
Then
we
add
to
∇
P
the
End(P)-valued
logarithmic
differential
given
by
log
log
∼
P
→
Ω
log
S/A
=
(0
⊕
O
S
)
⊗
O
S
Ω
S/A
⊆
P
⊗
O
S
Ω
S/A
where
the
first
morphism
is
the
projection
on
the
first
direct
summand.
The
resulting
logarithmic
connection
on
P
will
be
called
∇
P
.
Note
that
the
Kodaira-Spencer
morphism
for
F
1
(P)
⊆
P
with
respect
to
∇
P
is
the
identity
map.
Next,
we
define
the
Frobenius
action
on
P
as
follows:
We
take
the
Frobenius
action
to
be
the
morphism
P
Φ
:
Φ
∗
P
→
P
which
is
diagonal
with
respect
to
the
direct
sum
decomposition
P
=
Ω
log
S/A
⊕
O
S
and
is
equal
to
Φ
−1
on
O
S
and
to
dΦ
log
on
Ω
log
S/A
.
One
sees
easily
that
this
Frobenius
action
P
Φ
is
horizontal
with
respect
to
∇
P
.
Note
that
this
implies
that
∇
P
is
integrable,
since
its
∨
2
log
log
acts
curvature
would
define
a
Frobenius-invariant
section
of
(Ω
log
S/A
)
⊗
O
S
∧
Ω
S/A
,
but
Φ
on
this
bundle
with
slope
1,
so
any
Frobenius-invariant
section
must
necessarily
vanish.
Thus,
(P,
F
·
(P),
∇
P
,
P
Φ
)
defines
an
MF
∇
-object
in
the
sense
of
[Falt],
§2.
Definition
1.3.
We
shall
call
this
MF
∇
-object
the
uniformizing
MF
∇
-object
on
S
log
associated
to
Φ
log
.
91
log
Now
choose
a
base
point
s
:
Spf(A)
→
S
that
avoids
D.
Let
Π
S
log
=
π
1
(S
K
,
s
K
),
where
K
is
the
quotient
field
of
A,
and
by
the
fundamental
group
of
“S
log
,”
we
mean
the
fundamental
group
of
the
open
formal
subscheme
which
is
the
complement
of
divisors
that
define
the
log
structure.
Then
by
the
theory
of
[Falt],
§2,
the
uniformizing
MF
∇
-object
on
S
log
defines
a
dual
crystalline
Galois
representation
P
et
of
Π
S
log
.
Definition
1.4.
We
shall
call
this
Galois
representation
the
uniformizing
Galois
repre-
sentation
on
S
log
associated
to
Φ
log
.
Note
that
O
S
∼
=
(0
⊕
O
S
)
⊆
P
is
stabilized
by
∇
P
and
P
Φ
,
and
thus
defines
an
MF
∇
-
subobject
of
the
uniformizing
object
which
is
equal
to
the
trivial
MF
∇
-object.
If
we
take
the
quotient
of
the
uniformizing
MF
∇
-object
by
this
subobject,
we
obtain
the
MF
∇
-
object
corresponding
to
the
étale
Galois
representation
Θ
et
Φ
,
Tate
twisted
once.
Thus,
we
have
an
exact
sequence
of
Π
S
log
-modules
0
→
Θ
et
Φ
(1)
→
P
et
→
Z
p
→
0
which
thus
defines
an
extension
class:
η
Φ
∈
H
1
(Π
S
log
,
Θ
et
Φ
(1))
We
remark
relative
to
the
analogy
between
Frobenius
liftings
and
Kähler
metrics,
that
the
class
η
Φ
formally
“looks”
somewhat
like
the
differential
form
that
defines
a
Kähler
metric.
Also,
we
can
define
a
ring
with
Galois
action
which
will
be
useful
later.
First
consider
the
symmetric
algebra
on
Ω
et
Φ
(−1)
over
Z
p
:
S
·
Z
p
Ω
et
Φ
(−1)
Let
us
consider
the
Galois
action
of
Π
S
log
on
this
symmetric
algebra
which
differs
from
the
direct
sum
of
the
actions
on
the
S
i
(Ω
et
Φ
(−1))
by
the
class
η
Φ
.
Thus,
in
other
words,
Spec
of
the
symmetric
algebra
with
this
Galois
action
parametrizes
sections
of
the
exact
sequence
0
→
Θ
et
Φ
(1)
→
P
et
→
Z
p
→
0
If
we
then
adjoin
the
divided
powers
of
the
augmentation
ideal
to
this
Z
p
[Π
S
log
]-algebra,
we
obtain
a
Z
p
-algebra
A
Φ
.
Let
T
log
→
S
log
be
the
finite
covering
given
by
Φ
log
(so
92
T
log
∼
=
S
log
).
Since
this
finite
covering
is
log
étale
in
characteristic
zero,
we
may
form
the
subgroup
Π
T
log
⊆
Π
S
log
corresponding
to
this
covering.
Then
one
sees
easily
that
we
have
a
natural
Π
T
log
-action
on
A
Φ
,
compatible
with
the
Π
S
log
-action
on
the
symmetric
algebra.
(We
need
to
restrict
to
Π
T
log
so
that
the
Galois
action
respects
the
divided
powers.)
Definition
1.5.
We
shall
call
η
Φ
the
canonical
Galois
extension
class
associated
to
Φ
log
.
We
shall
call
A
Φ
the
ring
of
additive
periods
of
Φ
log
.
The
Canonical
p-divisible
Group
Let
us
look
at
the
uniformizing
MF
∇
-object
(P,
F
·
(P),
∇
P
,
P
Φ
)
again.
Let
U
⊆
S
be
the
open
formal
subscheme
which
is
the
complement
of
the
divisor
U
.
Then
by
[Falt],
Theorem
7.1,
this
MF
∇
-object
defines
a
p-divisible
group
G
Φ
over
U
.
Moreover,
just
as
with
the
corresponding
Galois
representation
discussed
in
the
previous
subsection,
we
have
an
exact
sequence
of
p-divisible
groups
over
U
:
0
→
Θ
et
Φ
(1)
⊗
Z
p
Q
p
/Z
p
→
G
Φ
→
Q
p
/Z
p
→
0
Definition
1.6.
We
shall
call
G
Φ
the
canonical
p-divisible
group
associated
to
Φ
log
.
Now
let
ω
∈
Ω
et
Φ
be
an
element
whose
reduction
modulo
p
is
nonzero.
Thus,
ω
defines
a
Z
p
-linear
morphism
ω(−)
:
Θ
et
Φ
→
Z
p
Let
S
n
→
S
be
the
étale
covering
defined
by
taking
the
kernel
of
the
morphism
Π
S
log
→
n
n
GL(Θ
et
Φ
⊗
Z/p
Z).
Let
U
n
→
U
be
its
restriction
to
U
.
Then
over
U
n
,
ω(−)
⊗
Z/p
Z
will
be
Galois
equivariant,
so
that,
by
pushing
forward
the
above
exact
sequence
by
means
of
ω(−),
we
obtain
an
exact
sequence
of
finite
flat
group
schemes:
0
→
Z/p
n
Z(1)
→
G
ω,n
→
Z/p
n
Z
→
0
which,
by
Kummer
theory,
defines
an
element
n
×
×
p
u
ω,n
∈
Γ(U
n
,
O
U
)/Γ(U
n
,
O
U
)
n
n
and
thus
a
differential
93
ω
n
=
(du
ω,n
)/u
ω,n
∈
Γ(U
n
,
Ω
U
n
/A
⊗
Z/p
n
Z)
def
Now
let
n
→
∞.
Let
S
be
the
p-adic
completion
of
the
inverse
limit
of
the
S
n
.
Since
the
various
ω
n
are
compatible,
we
thus
obtain
a
differential
,
Ω
ω
∈
Γ(
U
/A
)
U
Now
we
would
like
to
claim
that
ω
is
none
other
than
the
original
differential
ω
that
we
started
out
with.
In
some
sense,
I
believe
that
this
fact
is
well-known,
but
I
do
not
know
of
a
clear
reference
for
this
fact,
so
I
will
prove
it
explicitly
here.
First,
however,
we
need
to
make
a
few
more
general
observations
concerning
G
Φ
.
The
proof
will
be
given
in
the
subsection
after
the
next.
Logarithms
of
Periods
Suppose
that
k
is
algebraically
closed,
and
let
z
:
Spf(A)
→
S
be
a
rational
point
whose
reduction
modulo
p
is
equal
to
the
base
point
s.
In
particular,
it
follows
that
z
(since
it
coincides
with
the
base
point
s
maps
into
U
,
and
factors
canonically
through
U
modulo
p).
Thus,
we
can
restrict
the
G
ω,n
to
Spec(A)
via
z
so
as
to
obtain
an
extension
0
→
Q
p
/Z
p
(1)
→
G
ω,z
→
Q
p
/Z
p
→
0
of
p-divisible
groups
over
Spec(A).
By
Kummer
theory,
this
extension
defines
a
unit
u
ω,z
∈
A
×
whose
image
in
the
residue
field
k
is
1.
On
the
other
hand,
we
can
consider
the
Dieudonné
crystal
E
ω
of
G
ω,z
.
Thus,
E
ω
is
a
free
A-module
of
rank
two
with
a
filtration
F
1
(E
ω
)
⊆
E
ω
,
and
a
Frobenius
action
Φ
E
:
E
ω
F
→
E
ω
.
This
Frobenius
action
has
a
unique
subspace
E
F
⊆
E
ω
(respectively,
E
V
⊆
E
ω
)
on
which
Frobenius
acts
with
slope
zero
(respectively,
one).
Also,
E
V
and
F
1
(E
ω
)
define
the
same
subspace
modulo
p.
Since
F
1
(E
ω
)
and
E/F
1
(E
ω
)
are
naturally
isomorphic
to
A,
in
the
future,
we
shall
identify
them
with
A.
Thus,
by
projection
E
F
→
E
ω
→
E
ω
/F
1
(E
ω
)
=
A,
we
obtain
a
natural
isomorphism
of
E
F
with
A,
and,
dually,
a
natural
isomorphism
of
E
V
with
A.
Finally,
since
E
ω
=
E
F
⊕
E
V
,
we
may
regard
F
1
(E
ω
)
⊆
E
ω
as
the
graph
of
an
A-linear
morphism
A
=
E
V
→
A
=
E
F
,
which,
by
means
of
the
various
canonical
trivializations,
gives
us
an
element
L
ω,z
∈
p
·
A.
Theorem
1.7.
We
have
L
ω,z
=
log(u
ω,z
).
94
Proof.
Let
us
denote
the
sequence
of
Galois
modules
which
are
the
p-adic
Tate
modules
of
the
above
exact
sequence
of
p-divisible
groups
by
0
→
W
0
→
W
→
W
1
→
0.
Recall
the
exponential
map
of
[BK],
p.
359,
Definition
3.10,
1
exp
:
F
1
(E
ω
)
⊗−2
∼
(Z
p
(1))
=
A
→
A
×
=
H
Gal
1
where
the
first
isomorphism
is
the
trivialization
referred
to
above,
and
H
Gal
denotes
Galois
cohomology
with
respect
to
Gal(K/K),
where
K
is
the
quotient
field
of
K.
By
[BK],
p.
359,
Example
3.10.1,
one
knows
that
this
exponential
map
is
equal
to
the
ordinary
exponential
map
defined
by
the
exponential
series.
Let
η
2
=
log(u
ω,z
);
η
1
=
exp(η
2
).
Now
we
diagram-chase.
Let
us
denote
by
P
the
(infinite
dimensional)
Galois
module
B
crys
f
=1
⊕
B
DR
+
(notation
of
[BK]).
Applying
the
exact
sequence
(1.17.1)
of
[BK]
to
the
exact
sequence
of
Galois
modules
0
→
W
0
→
W
→
W
1
→
0,
we
obtain
the
following
commutative
diagram:
0
(W
0
)
H
Gal
⏐
⏐
−→
0
H
Gal
(W
)
⏐
⏐
−→
0
H
Gal
(W
1
)
⏐
⏐
−→
1
H
Gal
(W
0
)
⏐
⏐
0
(W
0
⊗
P)
H
Gal
⏐
⏐
−→
0
H
Gal
(W
⊗
P)
⏐
⏐
−→
0
H
Gal
(W
1
⊗
P)
⏐
⏐
−→
1
H
Gal
(W
0
⊗
P)
⏐
⏐
0
H
Gal
(W
0
⊗
B
DR
)
−→
0
H
Gal
(W
⊗
B
DR
)
0
−→
H
Gal
(W
1
⊗
B
DR
)
−→
1
H
Gal
(W
0
⊗
B
DR
)
0
(W
1
)
which
maps
via
the
connecting
homomorphism
to
Now
we
have
an
element
1
∈
H
Gal
1
0
1
η
1
∈
H
Gal
(W
);
since
the
image
of
η
1
in
H
Gal
(W
0
⊗P)
is
zero,
we
can
consider
log(η
1
)
=
η
2
.
0
0
On
the
other
hand,
1
∈
H
Gal
(W
1
)
maps
to
an
element
η
3
∈
H
Gal
(W
1
⊗
P)
that
dies
when
1
(W
0
⊗
P).
Thus
we
see
that
η
3
comes
from
hit
with
the
connecting
homomorphism
to
H
Gal
0
0
an
element
η
4
∈
H
Gal
(W
⊗
P)
which
is
unique
modulo
H
Gal
(W
0
⊗
P).
Mapping
η
4
down
0
0
0
one
step
to
H
Gal
(W
⊗
B
DR
),
we
get
η
5
∈
H
Gal
(W
⊗
B
DR
)
that
dies
in
H
Gal
(W
1
⊗
B
DR
),
0
0
0
0
hence
comes
from
a
unique
η
6
∈
H
Gal
(W
⊗
B
DR
)/H
Gal
(W
⊗
P)
=
T
V
.
Now
it
follows
from
the
explicit
definitions
of
the
maps
in
the
sequence
(1.17.1)
of
[BK]
that
η
6
is
precisely
L
ω,z
.
On
the
other
hand,
it
follows
from
general
principles
of
homological
algebra
that
η
6
=
η
2
.
This
completes
the
proof.
Compatibility
of
Differentials
Now
we
return
to
the
issue
of
showing
that
ω
=
ω.
Let
us
begin
by
observing
that
ω
,
we
can
also
be
defined
as
follows.
By
taking
the
direct
limit
of
the
G
ω,n
’s
restricted
to
U
obtain
an
extension
of
p-divisible
groups
0
→
Q
p
/Z
p
(1)
→
G
ω
→
Q
p
/Z
p
→
0
95
.
This,
moreover,
defines
a
Dieudonné
crystal
(E,
∇
E
)
with
a
filtration
F
1
(E)
⊆
E,
over
U
and
Frobenius
action.
In
fact,
(E,
∇
E
)
is
obtained
from
(P,
∇
P
)
simply
by
pulling
back
0
→
O
S
→
P
→
Ω
S/A
→
0
via
ω·
:
O
U
→
Ω
S/A
|
U
.
Now
F
1
(E)
and
E/F
1
(E)
may
be
identified
with
O
U
.
Thus,
ω
is
precisely
the
differential
obtained
by
considering
the
Kodaira-Spencer
morphism
F
1
(E)
=
O
U
→
Ω
U
/A
⊗
(E/F
1
(E))
=
Ω
U
/A
Now
let
R
z
be
the
complete
local
ring
which
is
the
completion
at
z
of
U
.
Let
R
z
PD
be
the
p-adic
completion
of
the
PD-envelope
of
R
z
at
the
augmentation
ideal
R
z
→
A
defined
by
z.
Now
taking
the
inverse
limit
of
the
u
ω,n
’s
defines
a
unit
u
ω
∈
(R
z
PD
)
×
whose
image
in
the
residue
field
k
is
1.
Thus,
we
can
consider
log(u
ω
)
∈
R
z
PD
.
On
the
other
hand,
let
P
z
=
P(z
∗
E)
=
P(E
ω
).
Let
σ
F
:
S
→
P
z
(respectively,
σ
V
:
S
→
P
z
)
denote
the
section
determined
by
the
subspace
E
F
⊆
E
ω
(respectively,
E
V
⊆
E
ω
).
The
trivializations
discussed
previously
define
an
isomorphism
of
the
tangent
space
to
P
z
at
σ
F
with
A.
Thus,
in
summary,
we
get
an
isomorphism
ψ
:
P
z
∼
=
P
1
by
sending
σ
F
(respectively,
σ
V
)
to
infinity
(respectively,
zero)
and
using
the
trivialization
of
tangent
space
to
σ
F
to
remove
the
remaining
multiplicative
ambiguity.
Let
P
R
z
PD
=
P(E)|
Spf(R
z
PD
)
.
Then
∇
E
gives
an
isomorphism
Ξ
P
:
P
R
z
PD
∼
=
R
z
PD
⊗
A
P
z
which,
when
composed
with
ψ,
gives
an
isomorphism
μ
:
P
R
z
PD
∼
=
P
1
R
z
PD
.
Now
by
Theorem
1.7,
it
follows
that
the
Hodge
section
(defined
by
F
1
(E)
→
E)
σ
:
Spf(R
z
PD
)
→
P
R
z
PD
∼
=
P
1
R
z
PD
is
(in
terms
of
the
standard
coordinate
t
on
P
1
,
which
vanishes
at
zero
and
has
a
pole
at
infinity)
simply
log(u
ω
).
(Indeed,
Theorem
1.7
tells
us
that
this
is
true
after
restriction
to
any
A-valued
point
of
R
z
PD
;
hence
it
must
be
true
over
R
z
PD
.)
It
thus
follows
that
the
def
pull-back
of
the
differential
dt
on
P
1
via
σ
is
simply
ω
=
du
ω
/u
ω
.
But,
tracing
through
all
the
definitions,
the
pull-back
of
dt
via
the
Hodge
section
is
exactly
the
Kodaira-Spencer
morphism
of
the
Hodge
filtration.
Thus,
we
conclude
that
ω
=
ω
over
R
z
PD
.
On
the
other
(since
O
→
R
PD
is
injective).
hand,
it
is
clear
that
this
implies
that
ω
=
ω
over
all
of
U
z
U
Thus,
we
have
proven
the
following
.
Theorem
1.8.
We
have
ω
=
du
ω
/u
ω
=
ω
over
U
96
Note
that
this
holds
(by
descent)
without
the
assumption
that
k
is
algebraically
closed.
Canonical
Liftings
of
Points
in
Characteristic
p
Let
α
1
∈
S(A)
be
an
A-valued
point
of
S.
Suppose
we
apply
Φ
to
α
1
to
obtain
an
A-valued
point
β
1
∈
S(A).
Then
since
Φ
is
a
Frobenius
lifting,
it
induces
zero
on
the
morphism
on
cotangent
spaces
modulo
p.
Thus,
β
1
(mod
p
2
)
depends
only
on
α
1
(mod
p).
Let
α
2
=
Φ
−1
A
(β
1
).
Thus,
α
2
≡
α
1
(mod
p),
and
α
2
depends
only
on
α
1
(mod
p).
If
we
then
continue
in
this
fashion,
defining
α
i+1
=
Φ
−1
A
Φ(α
i
)
def
it
is
clear
that
α
i
≡
α
1
(mod
p)
for
all
i
≥
1,
and
that
the
sequence
{α
i
}
of
points
in
S(A)
converges
p-adically.
Let
α
∞
∈
S(A)
be
the
limit
of
this
sequence.
Let
α
0
∈
S(k)
be
the
reduction
of
α
1
modulo
p.
Note
that
we
have
F
Φ(α
∞
)
=
α
∞
and
that,
moreover,
α
∞
is
the
unique
A-valued
point
of
S
which
has
this
property
and
is
equal
to
α
0
modulo
p.
Definition
1.9.
We
shall
call
α
∞
the
canonical
lifting
of
α
0
.
We
shall
call
an
A-valued
point
of
S
which
is
a
canonical
lifting
of
some
k-valued
point
a
canonical
A-valued
point
of
S.
All
the
canonical
extensions
that
we
have
defined
become
trivial
when
restricted
to
α
∞
.
More
precisely,
Proposition
1.10.
If
α
0
∈
U
(k),
then
the
restriction
of
u
ω
to
α
∞
is
1.
Proof.
Indeed,
the
Hodge
filtration
of
P
is
invariant
under
Φ,
so
its
restriction
to
α
∞
is
still
Frobenius
invariant.
By
the
theory
of
filtered
Dieudonné
modules
with
Frobenius
action
over
A
=
W
(k),
it
thus
follows
that
the
extension
of
p-divisible
groups
that
one
obtains
is
trivial.
Thus,
by
Kummer
theory,
u
ω
|
α
∞
=
1.
Canonical
Multiplicative
Parameters
Let
us
assume
just
in
this
subsection
that
k
is
algebraically
closed.
Let
z
∈
S(k)
be
a
k-valued
point
of
S.
Let
S
z
log
be
the
completion
of
S
log
at
z.
Thus,
S
z
log
is
Spf
of
a
complete
local
ring
R
z
which
is
noncanonically
isomorphic
to
97
A[[t
1
,
.
.
.
,
t
d
]]
with
the
restriction
of
the
divisor
D
defined
by
t
1
·
t
2
·
.
.
.
·
t
i
(where
i
may
be
zero).
Then
log
if
we
restrict
Ω
et
Φ
to
S
z
,
we
obtain
the
trivial
local
system.
Let
ω
∈
Ω
et
Φ
have
integral
residues
at
all
the
irreducible
components
of
D,
and
nonzero
reduction
modulo
p.
Thus,
ω
defines
a
surjection
ω(−)
:
Θ
et
Φ
→
Z
p
If
we
apply
ω(−)
to
our
canonical
extension
of
p-divisible
groups
0
→
Θ
et
Φ
(1)
⊗
Z
p
Q
p
/Z
p
→
G
Φ
→
Q
p
/Z
p
→
0
def
we
obtain
an
extension
of
Q
p
/Z
p
by
Q
p
/Z
p
(1)
over
U
z
=
S
z
|
U
.
By
Kummer
theory,
we
thus
obtain
a
“logarithmic
unit”
q
ω,z
∈
R
z
[
1
1
,
.
.
.
,
]
×
t
1
t
i
which
is
well-defined
up
to
multiplication
by
a
Teichmüller
representative
of
an
element
of
k.
If
ω
has
residue
e
j
at
the
component
of
D
defined
by
t
j
,
then
the
valuation
of
q
ω,z
at
(t
j
)
is
equal
to
e
j
.
Indeed,
this
follows
from
the
formula
dq
ω,z
/q
ω,z
=
ω
(of
Theorem
1.8).
Next,
let
us
consider
Φ
−1
(q
ω,z
).
Since
Φ
−1
multiplies
dq
ω,z
/q
ω,z
=
ω,
as
well
as
the
p
,
for
some
canonical
extension
of
p-divisible
groups
by
p,
it
follows
that
Φ
−1
(q
ω,z
)
=
λ
·
q
ω,z
λ
∈
[k
×
]
(where
the
brackets
mean
“the
Teichmüller
representative
of”).
On
the
other
hand,
because
Φ
is
a
Frobenius
lifting,
reducing
modulo
p
shows
that
λ
=
1.
Thus,
we
have
that
p
Φ
−1
(q
ω,z
)
=
q
ω,z
Definition
1.11.
We
shall
call
q
ω,z
for
such
an
ω
a
canonical
multiplicative
parameter
associated
to
Φ
log
.
Canonical
Affine
Coordinates
In
this
subsection,
k
need
not
be
algebraically
closed.
Let
α
∈
U
(A)
be
canonical.
Let
A
α
be
the
p-adic
completion
of
the
PD-envelope
of
S
at
the
subscheme
Im(α).
Let
α
→
A
be
the
augmentation
that
defines
the
point
α.
Let
I
=
Ker(
α
).
The
α
:
A
A-algebra
structure,
together
with
the
augmentation
α
define
a
splitting
98
A
α
=
A
⊕
I
which
we
shall
call
the
augmentation
splitting
of
A
α
.
Note
that
we
have
an
A-linear
Frobenius
action
Φ
A
:
(A
α
)
F
→
A
α
induced
by
the
Frobenius
lifting
Φ.
Moreover,
Φ
A
preserves
the
augmentation
splitting,
as
well
as
the
ring
structure
of
A
α
.
Let
us
consider
the
slopes
of
this
Frobenius
action
Φ
A
.
Clearly,
Φ
A
acts
on
A⊕0
⊆
A
α
as
Φ
A
.
Next,
we
note
that
since
Φ
A
is
a
Frobenius
lifting,
it
maps
I
into
p
·
I.
Thus,
we
have
Φ
A
(I
[j]
)
⊆
p
j
·
I
[j]
where
the
superscript
in
brackets
denotes
the
divided
power.
By
the
definition
of
an
def
ordinary
Frobenius
lifting,
Ω
α
=
I/I
[2]
has
constant
slope
one.
Thus,
if
we
divide
Φ
A
restricted
to
Ω
α
by
p,
we
obtain
an
isomorphism
(Ω
α
)
F
→
Ω
α
Next,
let
us
consider
the
A-submodule
Ω
can
⊆
I
which
is
the
closure
(in
the
p-adic
topology)
of
the
intersection
of
the
images
of
(
1
p
·
Φ
A
|
I
)
N
(for
all
N
≥
1).
Since
I/I
[2]
has
slope
1,
it
is
clear
that
the
projection
Ω
can
→
Ω
α
is
surjective.
Now
let
us
consider
the
intersection
of
Ω
can
with
I
[2]
.
Let
φ
=
(
p
1
·
Φ
A
|
I
)
N
(ψ),
where
ψ
∈
I.
If
φ
is
contained
in
I
[2]
modulo
p
N
,
then
since
I/I
[2]
has
slope
1,
it
follows
that
ψ
must
also
be
contained
in
I
[2]
modulo
p
N
.
But
then
φ
=
(
p
1
·Φ
A
|
I
)
N
(ψ)
must
be
zero
modulo
p
N
.
Thus,
we
conclude
that
the
projection
Ω
can
→
Ω
α
must
be
an
isomorphism.
Inverting
this
isomorphism,
we
thus
get
a
canonical
morphism
κ
A
:
Ω
α
→
A
α
Let
S
α
be
the
formal
scheme
which
is
the
p-adic
completion
of
the
PD-envelope
in
S
of
the
image
of
α
∈
U
(A).
Let
Θ
α
be
the
dual
A-module
to
Ω
α
.
Let
Θ
aff
α
be
the
p-adic
completion
of
the
PD-envelope
at
the
origin
of
the
affine
space
modeled
on
Θ
α
.
Thus,
Θ
aff
α
is
Spf
of
the
p-adic
completion
of
the
PD-envelope
of
the
symmetric
algebra
(over
A)
of
Ω
α
at
the
augmentation
ideal.
We
may
then
reinterpret
the
canonical
morphism
κ
A
as
an
isomorphism
α
U
can
:
Θ
aff
α
→
S
99
We
thus
see
that
we
have
proven
the
following
result:
Theorem
1.12.
For
every
choice
of
a
canonical
α
∈
U
(A),
we
obtain
a
local
uniformiza-
tion
(canonically
associated
to
Φ)
∼
α
U
can
:
Θ
aff
α
=
S
of
S
by
the
affine
space
modeled
on
Θ
α
.
Definition
1.13.
We
shall
call
the
elements
of
the
image
of
κ
A
canonical
affine
parameters
associated
to
Φ
at
α.
Now
let
(B,
m
B
)
be
a
local
ring
with
residue
field
k
which
is
p-adically
complete
and
has
a
topologically
nilpotent
PD-structure
on
m
B
.
Let
β
∈
S(B)
be
such
that
β(mod
m
B
)
∈
S(k)
is
equal
to
α(mod
p)
∈
S(k).
Let
S
β
be
the
p-adic
completion
of
the
PD-envelope
of
S
⊗
A
B
at
the
image
of
β
in
S(B).
Thus,
S
β
∼
=
S
α
⊗
A
B.
Let
B
β
=
A
α
⊗
A
B.
By
tensoring
the
canonical
morphism
κ
A
constructed
above
with
B,
we
thus
obtain
a
morphism
def
κ
B
:
(Ω
α
)
B
=
(Ω
α
)
⊗
A
B
→
B
β
Let
β
:
B
β
→
B
be
the
augmentation
corresponding
to
the
point
β
∈
S(B).
Let
I
β
=
Ker(β).
Let
B
β
=
B
⊕
I
β
be
the
splitting
defined
by
β
and
the
B-algebra
structure
on
B
β
.
Let
us
consider
the
projections
of
κ
B
on
these
two
factors:
κ
0
B
:
(Ω
α
)
B
→
B;
κ
1
B
:
(Ω
α
)
B
→
I
β
[2]
Let
Ω
β
=
I
β
/I
β
.
If
we
compose
κ
1
B
with
the
projection
to
Ω
β
,
we
thus
obtain
a
morphism
Ψ
αβ
:
(Ω
α
)
B
→
Ω
β
which
is
an
isomorphism,
since
it
is
an
isomorphism
modulo
m
B
,
where
β
coincides
with
α(mod
p).
Let
Θ
β
the
dual
B-module
to
Ω
β
.
Now
we
may
regard
κ
0
B
as
an
element
−1
of
(Θ
α
)
B
;
if
we
apply
(Ψ
∨
,
then
we
get
an
element
of
κ
β
∈
Θ
β
,
hence
∈
m
B
·
Θ
β
αβ
)
(since
β
≡
(
α
)
B
modulo
m
B
).
On
the
other
hand,
if
we
compose
κ
1
B
with
Ψ
−1
αβ
,
we
get
a
β
morphism
Ω
β
→
B
which
gives
us
an
isomorphism
100
β
U
β
:
Θ
aff
β
→
S
In
summary,
we
have
proven
the
following
result:
Theorem
1.14.
For
every
β
∈
S(B)
as
above,
we
obtain
a
canonical
class
κ
β
∈
m
B
·
Θ
β
,
as
well
as
a
local
uniformization
∼
β
U
β
:
Θ
aff
β
=
S
of
S
by
the
affine
space
modeled
on
Θ
β
.
Moreover,
this
uniformization
is
related
to
the
canonical
uniformization
by
tensoring
over
A
with
B,
applying
the
isomorphism
−1
:
(Θ
α
)
B
→
Θ
β
(Ψ
∨
αβ
)
and
then
translating
by
κ
β
.
Finally,
for
all
β
∈
S(B)
whose
reductions
modulo
m
B
are
equal
to
α(mod
p),
the
correspondence
β
→
κ
β
is
a
bijection
of
such
β
onto
m
B
·
Θ
β
.
Proof.
All
the
statements
except
the
last
follow
from
the
way
we
constructed
the
ob-
jects
involved.
The
last
statement
about
the
bijection
follows
from
simply
evaluating
the
canonical
uniformization
of
Theorem
1.12
on
B-valued
points.
Finally,
we
remark
(relative
to
the
analogy
between
Frobenius
liftings
and
Kähler
metrics)
that
these
canonical
affine
parameters
are
like
canonical
coordinates
for
a
real
analytic
Kähler
metric.
The
Relationship
Between
Affine
and
Multiplicative
Parameters
Let
us
continue
with
the
notation
of
the
preceding
subsection,
but
let
us
assume
in
addition
that
k
is
algebraically
closed,
so
that
the
canonical
multiplicative
parameters
are
defined.
Let
ω
∈
Ω
et
Φ
have
integral
residues
at
all
the
irreducible
components
of
D,
and
nonzero
reduction
modulo
p.
Since
Ω
et
Φ
⊆
Ω
α
,
we
may
also
regard
ω
as
an
element
of
Ω
α
.
Then
we
would
like
to
establish
the
relationship
between
the
canonical
multiplicative
parameter
q
ω,α
and
the
canonical
affine
parameter
κ
A
(ω).
First
note
that
by
Proposition
1.10,
q
ω,α
evaluated
at
α
is
a
Teichmüller
representative.
Thus,
log(q
ω,α
)
∈
A
α
is
zero
at
α,
as
is
κ
A
(ω)
∈
A
α
.
Moreover,
d
log(q
ω,α
)
=
ω,
by
Theorem
1.8.
On
the
other
hand,
the
fact
that
d
κ
A
(ω)
=
ω
is
a
tautology.
Thus,
log(q
ω,α
)
and
κ
A
(ω)
have
the
same
derivative
and
both
vanish
at
α.
We
thus
obtain
the
following
result:
101
Theorem
1.15.
We
have
κ
A
(ω)
=
log(q
ω,α
)
in
A
α
.
§2.
Construction
of
the
Canonical
Frobenius
Lifting
log
In
this
Chapter,
we
shall
denote
by
M
g,r
the
p-adic
formal
stack
of
r-pointed
stable
curves
of
genus
g
over
Z
p
.
We
shall
denote
the
reductions
of
objects
over
Z
p
to
F
p
or
Z/p
2
Z
ord
by
means
of
a
subscripted
F
p
or
Z/p
2
Z.
Let
(N
g,r
)
F
p
→
(M
g,r
)
F
p
be
the
étale
morphism
ord
in
characteristic
p
of
Chapter
II,
§3.
Thus,
(N
g,r
)
F
p
parametrizes
pairs
consisting
of
an
r-pointed
stable
curve
of
genus
g
in
characteristic
p,
together
with
an
ordinary,
nilpotent
ord
indigenous
bundle.
Let
N
g,r
→
M
g,r
be
the
unique
étale
morphism
of
p-adic
formal
ord
ord
schemes
that
lifts
(N
g,r
)
F
p
→
(M
g,r
)
F
p
.
Thus,
N
g,r
is
a
smooth
p-adic
formal
scheme
ord
over
Z
p
whose
reduction
modulo
p
is
(N
g,r
)
F
p
.
Our
goal
in
this
Section
is
to
construct
a
ord
canonical
ordinary
lifting
of
Frobenius
on
N
g,r
.
Modular
Frobenius
Liftings
In
this
subsection,
we
reinterpret
certain
constructions
from
Chapter
II,
§1,
in
terms
of
liftings
of
the
Frobenius
morphism
on
(M
g,r
)
F
p
.
Let
S
→
(M
g,r
)
F
p
be
étale,
and
let
S
log
log
be
the
log
scheme
obtained
by
pulling
back
the
log
structure
of
M
g,r
.
Thus,
in
particular,
S
is
smooth
over
F
p
.
Let
Φ
S
log
:
S
log
→
S
log
be
the
absolute
Frobenius
morphism.
Since
(M
g,r
)
F
p
⊆
(M
g,r
)
Z/p
2
Z
is
defined
by
a
nilpotent
ideal,
the
étale
morphism
S
→
(M
g,r
)
F
p
lifts
naturally
to
an
étale
morphism
S
→
(M
g,r
)
Z/p
2
Z
.
We
let
S
log
be
the
log
scheme
obtained
by
pulling
back
the
log
structure
of
(M
g,r
)
Z/p
2
Z
.
We
shall
call
a
Frobenius
lifting
on
S
log
a
log
morphism
S
log
→
S
log
whose
reduction
modulo
p
is
equal
to
Φ
S
log
.
Note
that
by
assigning
to
étale
morphisms
U
→
S
the
set
of
Frobenius
liftings
log
,
we
obtain
a
sheaf
L
on
the
étale
site
of
S,
with
the
natural
structure
of
a
torsor
on
U
def
∗
over
Θ
Φ
S
log
=
Φ
S
Θ
S
log
,
where
Θ
S
log
is
the
dual
vector
bundle
to
the
sheaf
of
logarithmic
differentials
on
S
log
.
Moreover,
this
torsor
naturally
admits
a
connection
∇
L
as
follows:
Consider
the
sheaf
of
bianalytic
functions
O
S
bi
on
S
log
.
The
image
I
Φ
S
of
the
Frobenius
−1
morphism
Φ
S
bi
:
O
S
bi
→
O
S
bi
is
equal
to
i
L
(Φ
−1
S
O
S
),
as
well
as
to
i
R
(Φ
S
O
S
)
(where
i
L
,
i
R
:
O
S
→
O
S
bi
are
the
left
and
right
injections).
Thus,
the
pull-back
of
the
sheaf
L
by
either
i
L
or
i
R
is
equal
to
the
sheaf
of
liftings
of
I
Φ
S
to
a
Z/p
2
Z-flat
subalgebra
of
O
S
bi
.
This
gives
a
connection
∇
L
on
the
Θ
Φ
S
log
-torsor
L
→
S
which
is
compatible
with
Φ
the
natural
connection
on
Θ
S
log
(for
which
sections
of
Φ
−1
S
Θ
S
log
are
horizontal);
also,
one
checks
easily
that
∇
L
is
integrable.
Now
let
us
recall
the
Θ
Φ
S
log
-torsor
D
→
S
that
we
defined
in
Chapter
II,
§1.
Let
log
f
log
:
X
log
→
S
log
be
the
pull-back
to
S
log
of
the
universal
curve
over
(M
g,r
)
F
p
.
Recall
102
log
F
then
that
D
is
the
Θ
Φ
)
=
X
log
×
S
log
,Φ
S
log
S
log
-torsor
consisting
of
liftings
of
the
curve
(X
S
log
→
S
log
to
an
S-flat
curve
Y
log
→
S
log
.
Note
that
it
follows
immediately
from
the
log
definition
of
the
classifying
log
stack
M
g,r
(plus
the
fact
that
S
→
M
g,r
is
étale)
that
we
have
an
isomorphism
α
:
D→L
log
→
S
log
(which
of
Θ
Φ
S
log
-torsors
given
by
considering
the
classifying
map
of
the
lifting
Y
is,
by
definition,
a
Frobenius
lifting
on
S
log
).
On
the
other
hand,
the
theory
of
Chap-
ter
II,
§1,
gives
a
natural
connection
∇
D
on
D
→
S
as
follows.
Recall
the
line
bundle
def
T
=
(Φ
X
log
/S
log
)
∗
(τ
X
log
/S
log
)
F
on
X
log
.
By
declaring
the
sections
of
the
τ
X
log
/S
log
inside
the
definition
of
T
to
be
horizontal,
we
see
that
T
gets
a
natural
connection
∇
T
over
X
log
(i.e.,
not
just
in
the
relative
sense
for
f
log
:
X
log
→
S
log
).
Thus,
the
de
Rham
cohomology
module
R
1
f
DR,∗
(T
)
has
a
Gauss-Manin
connection
∇
GM
on
S
log
.
By
Chapter
II,
Propo-
sition
1.1,
we
have
a
natural
surjection
R
1
f
DR,∗
(T
)
→
O
S
,
which
one
verifies
easily
to
be
horizontal.
Since,
by
Chapter
II,
Proposition
1.2,
D
is
just
the
sheaf
of
sections
of
this
surjection
R
1
f
DR,∗
(T
)
→
O
S
,
it
thus
follows
that,
as
such,
D
gets
a
natural
connection
(induced
by
∇
GM
),
which
we
shall
call
∇
D
.
Proposition
2.1.
The
isomorphism
α
is
horizontal
with
respect
to
∇
D
and
∇
L
.
Proof.
Let
us
denote
by
X
L
(respectively,
X
R
)
the
pull-back
of
f
log
:
X
log
→
S
log
via
i
L
:
O
S
→
O
S
bi
(respectively,
i
R
:
O
S
→
O
S
bi
)
to
S
bi
.
Thus,
we
obtain
a
diagram
over
S
bi
:
X
L
←
X
bi
→
X
R
Let
us
denote
the
left-pointing
(respectively,
right-pointing)
arrow
by
π
L
(respectively,
π
R
).
Similarly,
we
have
an
analogous
diagram
with
tildes,
for
the
various
objects
over
Z/p
2
Z.
Now
consider
the
image
of
Frobenius
I
Φ
S
⊆
O
S
bi
.
We
also
have
the
image
of
the
absolute
Frobenius
on
X,
which
we
denote
by
I
Φ
X
⊆
O
X
bi
.
Note
that
I
Φ
X
actually
sits
inside
both
O
X
L
and
O
X
R
.
Suppose
next
that
we
are
given
a
Z/p
2
Z-flat
subalgebra
I
Φ
S
⊆
O
S
bi
that
lifts
I
Φ
S
.
This
corresponds
to
a
section
η
of
L
L
=
L
R
.
(Here
the
superscript
“L”’s
and
“R”’s
denote
left
and
right
pull-backs
to
S
bi
,
respectively.)
The
obstruction
to
lifting
I
Φ
X
to
a
Z/p
2
Z-
1
L
L
S
flat
subalgebra
of
O
X
L
compatible
with
I
Φ
defines
a
class
in
R
(f
)
∗
(T
),
which
is,
by
def
def
definition,
equal
to
ξ[L]
=
(α
L
)
−1
(η).
Similarly,
we
obtain
a
class
ξ[R]
=
(α
R
)
−1
(η).
Note
that
(π
L
)
−1
(ξ[L])
=
(π
R
)
−1
(ξ[R]),
since
both
classes
are
the
obstruction
to
lifting
I
Φ
X
to
a
Z/p
2
Z-flat
subalgebra
of
O
X
bi
compatible
with
I
Φ
S
.
Let
us
call
this
common
class
ξ[bi].
103
Now
suppose
that
(
I
)
S
Φ
⊆
O
S
is
a
Z/p
2
Z-flat
lifting
of
I
Φ
S
.
Suppose
that
this
lift-
ing
corresponds
to
a
section
ζ
of
L.
If
we
then
take
η
=
ζ
L
(in
the
previous
para-
graph),
we
get
ξ[bi]
=
(π
L
)
−1
{(α
−1
(ζ))
L
},
and,
similarly,
if
we
take
η
=
ζ
R
,
we
get
ξ[bi]
=
(π
R
)
−1
{(α
−1
(ζ))
R
}.
On
the
one
hand,
∇
D
(α
−1
(ζ))
is
computed
by
subtracting
these
two
ξ[bi]’s.
On
the
other
hand,
(by
the
definition
of
the
Θ
Φ
S
log
-torsor
structure
on
D)
the
difference
between
these
two
ξ[bi]’s
is
the
difference
between
the
two
classifying
morphisms
given
by
the
subalgebras
{(
I
)
S
Φ
}
L
and
{(
I
)
S
Φ
}
R
of
O
S
bi
.
But
this
difference
is,
by
definition,
∇
L
(ζ).
This
completes
the
proof.
Henceforth,
we
shall
identify
(D,
∇
D
)
with
(L,
∇
L
),
and
call
the
resulting
torsor
with
connection
(D,
∇
D
)
(since
the
notation
L
is
more
natural
for
line
bundles).
Indigenous
Sections
of
D
We
continue
with
the
notation
of
the
previous
subsection.
Thus,
S
→
M
g,r
is
étale,
and
we
have
the
Θ
Φ
S
log
-torsor
D
→
S,
with
its
connection
∇
D
.
Let
π
:
D
→
S
be
the
scheme
corresponding
to
this
torsor.
Thus,
D
is
a
twisted
version
of
Spec
of
the
symmetric
algebra
of
the
dual
of
Θ
Φ
S
log
.
We
endow
D
with
the
log
structure
pulled
back
from
S;
this
log
gives
us
a
log
stack
D
.
On
D,
taking
the
dual
to
the
second
fundamental
exact
sequence
for
differentials
gives
a
sequence
of
tangent
bundles:
0
→
Θ
Φ
S
log
|
D
→
Θ
D
log
→
Θ
S
log
|
D
→
0
where
Θ
D
log
is
the
logarithmic
tangent
bundle
on
D
log
.
We
shall
denote
the
surjection
Θ
D
log
→
Θ
S
log
|
D
by
Θ
π
.
The
connection
∇
D
then
defines
a
connection
∇
D
on
the
fiber
bundle
π
:
D
→
S,
hence
a
section
∇
Θ
:
Θ
S
log
|
D
→
Θ
D
log
of
Θ
π
.
Now
let
us
suppose
that
we
are
given
a
section
σ
:
S
→
D
of
π.
Then
σ
induces
a
section
of
σ
∗
Θ
π
,
which
we
denote
by
Θ
σ
:
Θ
S
log
→
σ
∗
Θ
D
log
.
Also,
σ
defines
an
FL-bundle
(E,
∇
E
)
(see
Chapter
II,
§1)
on
the
curve
X
log
.
Definition
2.2.
We
shall
call
the
section
σ
indigenous
if
the
projectivization
of
the
FL-bundle
on
X
log
defined
by
σ
is
an
indigenous
bundle
on
X
log
.
Let
us
assume
that
σ
is
indigenous.
Then
we
obtain,
for
i
=
1,
2,
canonical
morphisms
of
vector
bundles
φ
i
:
σ
∗
Θ
D
log
→
Θ
S
log
defined
functorially
as
follows.
By
means
of
the
étale
morphism
S
→
M
g,r
,
we
can
think
of
the
geometric
vector
bundle
σ
∗
Θ
D
log
on
S
as
parametrizing
infinitesimal
deformations
η
=
{(X
log
)
,
(E
,
∇
E
)}
of
the
curve
plus
FL-bundle
pair
given
by
η
=
{X
log
,
(E,
∇
E
)}.
104
Then
the
obstruction
to
lifting
the
Hodge
filtration
of
(E
,
∇
E
)|
X
=
(E,
∇
E
)
(which
exists
since
σ
is
indigenous)
to
a
filtration
of
E
over
X
defines
a
section
of
Θ
S
log
,
which
we
take
for
φ
2
(η
).
On
the
other
hand,
if
we
think
in
terms
of
crystals,
then
(E
,
∇
E
)
also
defines
a
deformation
(E
,
∇
E
)
of
(E,
∇
E
)
on
X[
]/(
2
).
The
obstruction
to
lifting
the
Hodge
filtration
of
(E,
∇
E
)
to
a
filtration
of
E
on
X[
]/(
2
)
defines
a
section
of
Θ
S
log
,
which
we
take
for
φ
1
(η
).
Since,
σ
∗
Θ
π
(η
)
is
simply
the
difference
between
(X
log
)
and
the
trivial
deformation
of
X
log
,
we
thus
see
that
σ
∗
Θ
π
=
φ
2
−
φ
1
We
also
have
that
φ
1
◦
σ
∗
∇
Θ
=
0
Indeed,
sorting
through
the
definitions,
one
sees
that
the
image
of
∇
Θ
consists
of
the
η
=
{(X
log
)
,
(E
,
∇
E
)}
obtained
by
letting
(E
,
∇
E
)
be
the
FL-bundle
given
by
regarding
(E,
∇
E
)
as
a
crystal
and
taking
the
bundle
with
connection
that
this
crystal
induces
on
the
deformation
(X
log
)
.
Thus,
(E
,
∇
E
)
is
simply
the
trivial
deformation
of
(E,
∇
E
),
hence
is
indigenous
on
X
log
by
assumption.
Putting
the
above
two
formulas
together,
we
thus
obtain
that
φ
2
◦
σ
∗
∇
Θ
=
id
Θ
S
log
Also,
let
us
note
that
φ
2
◦
Θ
σ
=
0
since
if
it
were
nonzero,
it
would
measure
exactly
the
extent
to
which
σ
fails
to
stay
within
the
indigenous
locus
of
D,
but,
by
assumption,
σ
does
stay
within
the
indigenous
locus.
Next,
let
us
recall
the
morphism
Φ
τ
E
:
Θ
Φ
S
log
→
Θ
S
log
,
i.e.,
the
dual
to
the
“infinitesimal
Verschiebung”
of
Chapter
II,
§2.
Recall
that
this
morphism
was
constructed
by
applying
R
1
f
∗
to
the
morphism
T
→
τ
X
log
/S
log
given
by
composing
the
p-curvature
P
:
T
→
Ad(E)
of
E
with
the
projection
Ad(E)
→
τ
X
log
/S
log
arising
from
the
Hodge
filtration.
It
thus
follows
immediately
from
the
definitions
(by
thinking
about
how
one
defines
the
obstruction
∗
that
φ
1
measures)
that
if
we
restrict
φ
1
:
σ
∗
Θ
D
log
→
Θ
S
log
to
Θ
Φ
S
log
⊆
σ
Θ
D
log
,
we
get
φ
1
|
Θ
Φ
S
log
=
Φ
τ
E
So
far
we
have
been
thinking
about
morphisms
that
one
can
obtain
from
σ
by
thinking
about
the
indigenous
FL-bundle
(E,
∇
E
)
that
it
defines.
But
by
what
we
did
in
the
previous
subsection,
σ
also
defines
a
Frobenius
lifting
Φ
σ
:
S
log
→
S
log
.
Let
us
consider
the
105
morphism
Θ
Φ
σ
:
Θ
S
log
→
Θ
Φ
S
log
obtained
by
looking
at
the
morphism
induced
by
Φ
σ
on
the
tangent
bundles,
and
then
dividing
by
p.
On
the
other
hand,
the
morphism
Θ
σ
−σ
∗
∇
Θ
:
∗
Θ
S
log
→
σ
∗
Θ
D
log
maps
into
Θ
Φ
S
log
⊆
σ
Θ
D
log
.
Thus,
by
abuse
of
notation,
we
shall
regard
∗
Θ
σ
−
σ
∇
Θ
as
a
morphism
Θ
S
log
→
Θ
Φ
S
log
.
Then
we
claim
that
Θ
Φ
σ
=
Θ
σ
−
σ
∗
∇
Θ
Indeed,
if
we
think
of
σ
∗
∇
Θ
as
defining
a
direct
sum
splitting
of
σ
∗
Θ
D
log
,
then
Θ
σ
−
σ
∗
∇
Θ
∗
is
just
the
component
of
Θ
σ
that
sits
in
the
vertical
subspace
Θ
Φ
S
log
⊆
σ
Θ
D
log
.
Put
another
∗
Φ
way,
Θ
σ
−σ
∇
Θ
:
Θ
S
log
→
Θ
S
log
is
the
Kodaira-Spencer
morphism
for
the
section
σ
relative
to
the
connection
∇
D
.
Thus,
it
follows
from
the
definition
of
the
connection
called
∇
L
in
the
previous
subsection
in
terms
of
subalgebras
of
O
S
bi
that
Θ
Φ
σ
=
Θ
σ
−
σ
∗
∇
Θ
.
We
are
now
ready
to
prove
the
main
technical
result
of
this
subsection:
Proposition
2.3.
If
σ
:
S
→
D
is
an
indigenous
section,
then
−Φ
τ
E
is
inverse
to
Θ
Φ
σ
.
In
particular,
the
indigenous
bundle
associated
to
(E,
∇
E
)
is
ordinary.
Proof.
Indeed,
using
the
various
observations
made
above,
we
simply
compute:
Φ
τ
E
◦
Θ
Φ
σ
=
φ
1
◦
(Θ
σ
−
σ
∗
∇
Θ
)
=
φ
1
◦
Θ
σ
=
(φ
1
−
φ
2
)
◦
Θ
σ
=
−(σ
∗
Θ
π
)
◦
Θ
σ
=
−id
Θ
S
log
Thus,
in
particular
Φ
τ
E
is
an
isomorphism,
and
so
(E,
∇
E
)
defines
an
ordinary
nilpotent
indigenous
bundle.
Frobenius
Invariant
Indigenous
Bundles
ord
In
this
subsection,
we
change
notation
slightly.
Let
S
→
N
g,r
be
an
étale
morphism
ord
of
a
p-adic
formal
scheme
S
into
N
g,r
.
Thus,
S
is
formally
smooth
over
Z
p
.
Also,
one
may
ord
ord
think
of
S
→
N
g,r
as
the
unique
étale
lifting
of
its
reduction
S
F
p
→
(N
g,r
)
F
p
modulo
p.
For
convenience,
we
assume
that
S
F
p
is
affine.
Endow
S
with
the
log
structure
pulled
log
back
from
M
g,r
.
Thus,
we
get
a
p-adic
formal
log
scheme
S
log
.
Pulling
back
the
universal
curve
over
M
g,r
,
we
get
a
morphism
f
log
:
X
log
→
S
log
.
Let
h
log
:
Y
log
→
S
log
be
an
log
r-pointed
curve
of
genus
g
whose
reduction
modulo
p
is
equal
to
(X
log
)
F
F
p
→
S
F
p
,
i.e.,
the
Frobenius
transform
of
f
F
log
.
We
shall
denote
the
divisor
of
marked
points
on
Y
by
E
⊆
Y
.
p
106
Let
n
≥
2
be
a
natural
number.
Suppose
that
we
have
a
coherent
sheaf
with
connec-
tion
(F,
∇
F
)
on
Y
log
,
where
F
is
killed
by
p
n
and
flat
over
Z/p
n
Z,
and
the
connection
∇
F
is
relative
to
the
morphism
h
log
:
Y
log
→
S
log
.
Suppose,
moreover,
that
we
are
given
a
filtration
F
1
(F)
F
p
⊆
F
F
p
of
the
reduction
of
F
modulo
p.
We
shall
call
this
filtration
the
Hodge
filtration.
Then,
relative
to
this
data,
we
define
the
coherent
sheaf
with
con-
/S
log
).
nection
F
∗
(F,
∇
F
)
as
follows.
First,
we
regard
(F,
∇
F
)
as
a
crystal
on
Crys(Y
F
log
p
Thus,
if
we
apply
the
relative
Frobenius
morphism
Φ
X
log
/S
log
to
this
crystal
(F,
∇
F
),
we
def
obtain
a
crystal
(F,
∇
F
)
=
Φ
∗
X
log
/S
log
(F,
∇
F
)
on
Crys(X
F
log
/S
log
).
Next,
we
consider
p
the
subsheaf
Φ
∗
X
log
/S
log
F
1
(F)
F
p
⊆
F
F
p
.
If
we
then
consider
the
subsheaf
of
(F,
∇
F
)
on
Crys(X
F
log
/S
log
)
consisting
of
sections
whose
reduction
modulo
p
is
contained
in
the
p
subsheaf
Φ
∗
X
log
/S
log
F
1
(F)
F
p
,
we
obtain
a
crystal
(F,
∇
F
)
on
Crys(X
F
log
/S
log
).
We
then
p
let
F
∗
(E,
∇
E
)
=
(F,
∇
F
)
⊗
Z
p
Z/p
n−1
Z
def
Definition
2.4.
We
shall
call
F
∗
(F,
∇
F
)
the
renormalized
Frobenius
pull-back
of
(F,
∇
F
).
Note
that
if
F
is
a
vector
bundle
on
Y
Z/p
n
Z
,
and
F
1
(F)
F
p
is
a
vector
bundle
on
Y
F
p
,
with
the
injection
F
1
(F)
F
p
→
F
F
p
locally
split,
it
follows
immediately
from
the
definitions
that
the
coherent
sheaf
F
∗
(F)
appearing
in
F
∗
(F,
∇
F
)
is
a
vector
bundle
on
X
Z/p
n−1
Z
.
Note
also
that,
if
we
think
of
the
“input
variable”
(F,
∇
F
)
as
a
crystal
on
Crys(Y
F
log
/S
log
),
then
p
.
F
∗
(F,
∇
F
)
does
not
depend
on
the
choice
of
the
deformation
h
log
:
Y
log
→
S
log
of
Y
F
log
p
log
log
n
Often,
we
will
be
given
a
Frobenius
lifting
Φ
log
:
S
Z/p
n
Z
→
S
Z/p
n
Z
modulo
p
,
and
we
log
log
log
log
will
take
Y
Z/p
n
Z
→
S
Z/p
n
Z
(respectively,
(F,
∇
F
))
to
be
the
pull-back
of
X
Z/p
n
Z
→
S
Z/p
n
Z
(respectively,
some
(E,
∇
E
)
on
X
log
)
by
Φ
log
.
If
S
were
the
spectrum
of
the
ring
of
Witt
vectors
of
a
perfect
field
k,
and
n
=
2,
then
the
F
∗
(E,
∇
E
)
F
that
we
have
defined
here
would
coincide
with
the
F
∗
(E,
∇
E
)
of
Chapter
II,
Definition
2.9.
log
Now
let
(F,
∇
F
)
be
a
vector
bundle
with
connection
on
Y
Z/p
n
Z
whose
determinant
n−1
is
trivial
and
which
is
indigenous
modulo
p
.
We
will
denote
its
Hodge
filtration
by
F
1
(F)
Z/p
n−1
Z
⊆
F
Z/p
n−1
Z
.
Let
us
denote
by
(G,
∇
G
)
the
vector
bundle
with
connection
on
X
Z/p
n−1
Z
which
is
the
renormalized
Frobenius
pull-back
of
(F,
∇
F
).
Suppose,
more-
∼
over,
that
(G,
∇
G
)
F
F
p
=
(F,
∇
F
)
F
p
.
Thus,
one
sees
(as
in
the
proof
of
Chapter
II,
Propo-
sition
2.10)
that
(F,
∇
F
)
F
p
is
nilpotent
and
admissible
(hence
corresponds,
by
Chapter
II,
Proposition
2.5,
to
some
FL-bundle).
Lemma
2.5.
Let
n
≥
3.
If
we
modify
the
connection
∇
F
by
some
p
n−2
θ,
where
θ
is
a
section
of
h
∗
(ω
Y
log
/S
)
⊗2
(−E),
then
the
vector
bundle
F
∗
(F)
(on
X
Z/p
n−1
Z
),
along
with
its
connection
F
∗
(∇
F
),
remain
unchanged.
107
Proof.
Looking
at
the
definition
of
the
renormalized
Frobenius
pull-back,
one
sees
that
the
pair
(F
∗
(F),
F
∗
(∇
F
))
is
constructed
by
pulling
back
F
(and
∇
F
)
via
various
local
liftings
of
Φ
X
log
/S
log
,
and
then
gluing
together
by
means
of
gluing
morphisms
defined
by
the
connection
∇
F
.
Moreover,
these
gluing
morphisms
are
obtained
from
the
Taylor
expansion
(cf.
[Falt],
§2,
Theorem
2.3),
which
involves
applying
the
connection
∇
F
to
tangent
vectors
pushed
forward
from
the
Frobenius
lifting.
Since
such
tangent
vectors
are
necessarily
divisible
by
p
(as
well
as
being
annihilated,
of
course,
by
p
n
),
it
follows
that
a
knowledge
of
(∇
F
)
Z/p
n−1
Z
suffices
to
compute
these
Taylor
expansions.
Thus,
certainly
F
∗
(F)
depends
at
most
on
(∇
F
)
Z/p
n−1
Z
.
On
the
other
hand,
since
at
the
end
of
the
construction
of
F
∗
(F),
we
mod
out
by
p
n−1
·
Φ
∗
X
log
/S
log
F
1
(F)
F
p
⊆
p
n−1
·
Φ
∗
X
log
/S
log
F
F
p
,
we
see
that
modifying
∇
F
by
an
endomorphism-valued
differential
whose
image
lies
inside
p
n−2
·
F
1
(F)
(where
we
have
p
n−2
rather
than
p
n−1
since
we
always
get
an
extra
factor
of
p
from
the
fact
that
we
are
applying
the
connection
to
tangent
vectors
divisible
by
p)
does
not
affect
the
result.
This
completes
the
proof.
Now
let
us
assume
that
(F,
∇
F
)
is
a
rank
two
vector
bundle
on
Y
Z/p
n
Z
with
a
con-
nection
(relative
to
h
log
:
Y
log
→
S
log
),
whose
determinant
is
trivial.
Let
us
suppose,
moreover,
that
(F,
∇
F
)
Z/p
n−1
Z
is
indigenous.
Let
(G,
∇
G
)
=
F
∗
(F,
∇
F
).
As
before,
we
∗
∼
assume
that
(G,
∇
G
)
F
F
p
=
(F,
∇
F
)
F
p
.
Then
by
considering
the
result
of
applying
F
to
var-
ious
deformations
(F,
∇
F
)
of
(F,
∇
F
)
(i.e.,
such
that
(F,
∇
F
)
Z/p
n−1
Z
=
(F,
∇
F
)
Z/p
n−1
Z
)
to
obtain
various
deformations
(G,
∇
G
)
of
(G,
∇
G
),
we
obtain
a
morphism:
(R
1
f
DR,∗
Ad(E)
F
p
)
F
→
R
1
f
DR,∗
Ad(G)
F
p
∼
=
R
1
f
DR,∗
Ad(E)
F
p
If
we
then
compose
this
morphism
with
the
projection
R
1
f
DR,∗
Ad(E)
F
p
→
R
1
f
∗
(τ
X
log
/S
log
)
F
p
arising
from
the
Hodge
filtration,
we
obtain
a
morphism
(R
1
f
DR,∗
Ad(E)
F
p
)
F
→
R
1
f
∗
(τ
X
log
/S
log
)
F
p
which,
by
Lemma
2.5,
vanishes
on
the
subbundle
log
⊗2
1
F
)
(−D))
F
(f
∗
(ω
X/S
F
p
⊆
(R
f
DR,∗
Ad(E)
F
p
)
arising
from
the
Hodge
filtration.
Thus,
we
obtain
a
morphism
of
vector
bundles
Θ
F
∗
:
(R
1
f
∗
(τ
X
log
/S
log
)
F
p
)
F
→
R
1
f
∗
(τ
X
log
/S
log
)
F
p
Note
that
by
Lemma
2.5,
this
morphism
remains
unchanged
if
one
adds
some
p
n−2
θ
to
the
connection
∇
F
.
108
Lemma
2.6.
The
morphism
Θ
F
∗
is
equal
to
−Φ
τ
E
.
Proof.
In
the
gluing
process
referred
to
in
the
proof
of
Lemma
2.5,
the
deforming
n−1
cocycle
in
(τ
X
log
/S
log
)
F
(τ
X
log
/S
log
)
F
(mod
p
n
)
only
affects
the
Taylor
expan-
F
p
=
p
sion
to
first
order.
Moreover,
this
cocycle
in
(τ
X
log
/S
log
)
F
F
p
is
mapped
to
a
cocycle
in
∗
F
Φ
X
log
/S
log
(τ
X
log
/S
log
)
F
p
→
Ad(G)
F
p
∼
=
Ad(E)
F
p
,
and
hence
to
a
cocyle
in
Ad(E)
F
p
.
If
we
then
further
project
this
cocycle
via
Ad(E)
F
p
→
(τ
X
log
/S
log
)
F
p
,
we
obtain
Θ
F
∗
of
the
original
cocycle.
On
the
other
hand,
let
us
note
that
by
Chapter
II,
Proposition
1.4,
the
inclusion
Φ
∗
X
log
/S
log
(τ
X
log
/S
log
)
F
F
p
→
Ad(G)
F
p
=
Ad(E)
F
p
is
−1
times
the
p-curvature
of
τ
(E,
∇
E
)
F
p
.
Since
Φ
E
is
defined
by
applying
R
1
f
∗
to
the
p-curvature
composed
with
the
projection
Ad(E)
F
p
→
(τ
X
log
/S
log
)
F
p
,
we
thus
obtain
the
result.
We
are
now
ready
to
begin
defining
a
canonical
Frobenius
lifting
on
S
log
,
which
will
be
fundamental
to
the
entire
paper.
First,
note
that
since
(N
g,r
)
F
p
⊆
(S
g,r
)
F
p
,
we
have
a
ord
tautological
trivialization
(τ
N
)
F
p
:
(N
g,r
)
ord
F
p
→
(S
g,r
)
F
p
of
the
torsor
S
g,r
over
(N
g,r
)
F
p
.
If
we
pull
this
trivialization
back
to
S,
we
get
a
trivialization
(τ
S
)
F
p
:
S
F
p
→
(S
g,r
)
F
p
,
.
This
indige-
which
thus
defines
a
nilpotent,
ordinary
indigenous
bundle
(E,
∇
E
)
1
on
X
F
log
p
nous
bundle
thus
corresponds
to
an
FL-bundle,
hence
a
section
of
the
torsor
D
(of
the
log
log
previous
subsection)
over
S
F
p
,
and
hence
a
Frobenius
lifting
Φ
log
2
:
S
Z/p
2
Z
→
S
Z/p
2
Z
.
Now
log
let
(E,
∇
E
)
2
be
any
indigenous
bundle
on
X
Z/p
2
Z
that
lifts
(E,
∇
E
)
1
.
(Such
a
lifting
exists
log
log
log
since
S
F
p
is
affine.)
We
shall
define
Y
log
inductively.
Let
Y
Z/p
.
2
Z
=
X
Z/p
2
Z
×
S
log
,Φ
log
S
2
∗
∗
Let
(F,
∇
F
)
2
=
(Φ
log
2
)
(E,
∇
E
)
2
.
Then
it
is
a
tautology
that
if
we
take
F
(F,
∇
F
)
2
,
we
obtain
(E,
∇
E
)
1
(up
to
tensor
product
with
a
line
bundle
with
connection
whose
square
is
trivial;
as
usual,
for
the
sake
of
simplicity,
we
shall
ignore
this).
So
far,
to
summarize,
log
of
the
objects
constructed
so
far,
Φ
log
2
;
Y
Z/p
2
Z
;
and
(E,
∇
E
)
1
are
canonical.
The
primed
objects
are
not
canonical.
log
Let
(F,
∇
F
)
3
be
any
rank
two
bundle
with
connection
on
Y
Z/p
3
Z
whose
determinant
is
trivial,
and
whose
reduction
modulo
p
2
is
equal
to
(F,
∇
F
)
2
.
That
is,
(F,
∇
F
)
3
is
a
deformation
of
(F,
∇
F
)
2
.
Now
by
Lemma
2.6,
and
the
fact
that
(E,
∇
E
)
1
is
ordinary,
it
follows
that,
among
all
possible
deformations
(F,
∇
F
)
3
of
(F,
∇
F
)
2
,
there
exists
a
unique
(up
to
changing
the
connection
by
some
p
2
·
θ)
such
deformation
(F,
∇
F
)
3
such
that
log
log
log
F
∗
(F,
∇
F
)
3
is
indigenous
on
X
Z/p
2
Z
.
Let
Y
Z/p
3
Z
be
the
unique
deformation
of
Y
Z/p
2
Z
log
such
that
when
one
evaluates
the
crystal
(F,
∇
F
)
3
on
Y
Z/p
3
Z
,
it
becomes
indigenous.
∗
Let
(E,
∇
E
)
2
=
F
(F,
∇
F
)
3
.
By
Lemma
2.5,
(E,
∇
E
)
2
is
independent
of
the
choice
of
log
log
log
∗
:
S
Z/p
(E,
∇
E
)
2
or
(F,
∇
F
)
3
.
Let
(F,
∇
F
)
2
=
(Φ
log
3
Z
→
S
Z/p
3
Z
be
2
)
(E,
∇
E
)
2
.
Let
Φ
3
log
log
the
classifying
morphism
of
the
r-pointed
stable
curve
of
genus
g
given
by
Y
Z/p
3
Z
→
S
Z/p
3
Z
.
log
log
log
Thus,
Φ
log
3
lifts
Φ
2
.
Again,
to
summarize,
the
objects
Φ
3
;
Y
Z/p
3
Z
;
and
(E,
∇
E
)
2
(as
well
log
as
(F,
∇
F
)
2
)
are
canonical.
If
we
now
let
(E,
∇
E
)
3
be
an
indigenous
bundle
on
X
Z/p
3
Z
109
∗
that
lifts
(E,
∇
E
)
2
,
and
(F,
∇
F
)
3
=
(Φ
log
3
)
(E,
∇
E
)
3
(where
the
primed
bundles
are
newly
chosen
here,
hence
different
from
the
temporary
ones
we
chose
before),
it
follows
from
Lemma
2.5
that
(E,
∇
E
)
2
∼
=
F
∗
(F,
∇
F
)
3
.
Continuing
in
this
fashion
(making
repeated
use
of
Lemmas
2.5
and
2.6,
as
well
as
the
fact
that
(E,
∇
E
)
1
is
ordinary),
we
thus
obtain
a
canonical
Frobenius
lifting
Φ
log
:
S
log
→
S
log
(of
p-adic
formal
schemes),
as
well
as
a
canonical
indigenous
bundle
(E,
∇
E
)
on
X
log
such
that
F
∗
(Φ
log
)
∗
(E,
∇
E
)
∼
=
(E,
∇
E
)
(up
to
tensor
product
with
a
line
bundle
with
connection
whose
square
is
trivial).
Moreover,
note
that
by
Proposition
2.3,
this
Frobenius
lifting
Φ
log
is
ordinary.
Definition
2.7.
Let
Ψ
log
:
S
log
→
S
log
be
a
Frobenius
lifting.
We
shall
call
an
indigenous
bundle
(G,
∇
G
)
on
X
log
Frobenius
invariant
for
Ψ
log
if
(G,
∇
G
)
∼
=
F
∗
(Ψ
log
)
∗
(G,
∇
G
)
(up
to
tensor
product
with
a
line
bundle
with
connection
whose
square
is
trivial).
So
far,
we
have
been
working
over
our
affine
scheme
S,
which
is
étale
over
M
g,r
.
However,
since
the
objects
that
we
have
constructed
(namely,
Φ
log
and
(E,
∇
E
))
are
canonical,
i.e.
uniquely
characterized
by
certain
properties
that
have
nothing
special
to
do
with
S,
it
is
ord
clear
that
they
all
descend
to
(N
g,r
)
log
.
We
thus
see
that
we
have
proven
the
following
key
result:
ord
Theorem
2.8.
On
(N
g,r
)
log
,
there
exists
a
canonical
ordinary
Frobenius
lifting
ord
ord
log
Φ
log
→
(N
g,r
)
log
N
:
(N
g,r
)
ord
together
with
a
canonical
indigenous
bundle
(E
N
,
∇
E
N
)
on
C
log
(where
C
log
→
N
g,r
is
the
universal
r-pointed
curve
of
genus
g)
whose
reduction
modulo
p
is
equal
to
the
nilpotent,
ordinary
indigenous
bundle
defined
by
the
tautological
trivialization
(τ
N
)
F
p
of
S
g,r
over
ord
N
g,r
.
Moreover,
the
pair
{Φ
log
N
;
(E
N
,
∇
E
N
)}
is
uniquely
characterized
by
the
following
properties:
ord
ord
log
→
(N
g,r
)
log
is
a
lifting
of
Frobenius;
(1)
Φ
log
N
:
(N
g,r
)
(2)
the
reduction
of
(E
N
,
∇
E
N
)
modulo
p
is
the
bundle
defined
by
(τ
N
)
F
p
;
(3)
(E
N
,
∇
E
N
)
is
Frobenius
invariant
for
Φ
log
N
.
Moreover,
the
formation
of
Φ
log
N
and
(E
N
,
∇
E
N
)
is
compatible
with
restriction
to
products
ord
ord
of
N
g,r
’s
for
smaller
g’s
and
r’s
that
map
into
the
boundary
of
our
original
N
g,r
via
the
gluing
procedure
described
at
the
end
of
Chapter
I,
§2.
110
Proof.
We
have
proven
everything
except
the
last
statement
about
restriction.
To
see
ord
this,
note
first
of
all
that
Φ
log
N
respects
such
products
of
smaller
N
g,r
since
it
respects
the
ord
log
structure
of
the
original
N
g,r
.
Thus,
we
may
restrict
Φ
log
N
and
(E
N
,
∇
E
N
)
to
these
products,
and
the
result
follows
by
uniqueness.
Remark.
This
result
is
the
central
result
of
this
paper.
In
some
sense,
the
rest
of
the
paper
is
just
devoted
to
making
explicit
a
number
of
formal
consequences
of
Theorem
2.8.
In
particular,
since
this
canonical
Frobenius
lifting
is
ordinary,
it
follows
that
we
can
apply
the
theory
of
§1.
We
shall
proceed
to
do
this
in
the
remainder
of
this
Chapter.
Finally,
it
is
useful
to
know
that
the
formation
of
the
canonical
Frobenius
and
indige-
ord
nous
bundle
are
compatible
with
finite
coverings.
Suppose
that
S
log
→
(N
g,r
)
log
is
log
étale,
with
S
formally
smooth
over
Z
p
,
and
the
log
structure
given
by
a
relative
divisor
with
normal
crossings
over
Z
p
.
Let
f
log
:
X
log
→
S
log
be
the
pull-back
of
the
universal
log
curve
over
M
g,r
.
Let
q,
s
≥
0
be
such
that
2q
−
2
+
s
≥
1.
Let
Y
log
→
S
log
be
an
s-pointed
stable
curve
of
genus
q.
Suppose
that
we
are
given
a
morphism
over
S
log
:
Y
log
φ
log
−→
X
log
Now
we
make
the
following:
Definition
2.9.
We
shall
say
that
φ
log
is
log
admissible
if
it
is
finite,
log
étale,
and
takes
marked
points
to
marked
points.
A
typical
example
of
a
log
admissible
morphism
may
be
obtained
by
considering
the
“admissible
coverings”
of
[HM].
Indeed,
it
is
not
difficult
to
see
that
by
endowing
the
curves
involved
(as
well
as
the
base)
with
appropriate
log
structures,
one
may
obtain
a
log
admissible
covering
(cf.
[Mzk],
§3).
(Note,
however,
that
the
definition
of
“log
admissible”
given
here
differs
from
that
of
[Mzk],
§3.)
Let
(E,
∇
E
)
be
the
restriction
of
the
canonical
indigenous
bundle
(E
N
,
∇
E
N
)
to
X
log
.
log
(which
exists
Let
Φ
log
:
S
log
→
S
log
be
the
pull-back
of
the
Frobenius
lifting
Φ
log
N
to
S
ord
because
S
log
→
(N
g,r
)
log
is
log
étale).
Let
(F,
∇
F
)
=
φ
∗
(E,
∇
E
).
Observe
that
(F,
∇
F
)
F
p
is
a
nilpotent,
admissible
indigenous
bundle
on
Y
log
.
Let
us
assume
that:
(*)
(F,
∇
F
)
F
p
is
ordinary.
log
Then
(F,
∇
F
)
F
p
determines
a
factorization
of
the
classifying
morphism
S
log
→
M
q,s
ord
through
(N
q,s
)
log
.
Thus,
we
get
a
morphism
ord
κ
log
:
S
log
→
(N
q,s
)
log
111
ord
For
simplicity,
let
us
write
T
log
for
(N
q,s
)
log
.
Let
us
denote
by
Ψ
log
the
canonical
Frobenius
ord
on
(N
q,s
)
log
,
and
by
(G,
∇
G
)
the
canonical
indigenous
bundle
on
the
universal
s-pointed
ord
stable
curve
of
genus
q
over
(N
q,s
)
log
.
Then
we
have
the
following
compatibility
result:
Theorem
2.10.
We
have
a
commutative
diagram:
Φ
log
S
log
−→
⏐
⏐
log
κ
T
log
log
Ψ
−→
S
log
⏐
⏐
log
κ
T
log
and
an
isomorphism
κ
∗
(G,
∇
G
)
∼
=
(F,
∇
F
).
Proof.
We
shall
apply
induction
on
i
to
the
proposition
“the
Theorem
is
true
when
the
objects
in
it
are
reduced
modulo
p
i
.”
The
case
i
=
1
is
clear.
Thus,
it
suffices
to
prove
the
induction
step.
Let
us
consider
the
crystals
(F,
∇
F
)
Φ
and
κ
∗
(G,
∇
G
)
Ψ
on
Crys(X
F
log
/S
log
).
p
Suppose
that
they
agree
modulo
p
i
.
If
we
apply
F
∗
(the
renormalized
Frobenius
pull-
back)
to
them,
we
get
the
same
crystal
modulo
p
i
,
by
the
induction
hypothesis
and
the
definition
of
the
canonical
Frobenii
and
indigenous
bundles.
Thus,
by
Lemma
2.6,
it
follows
that
the
underlying
vector
bundles
of
(F,
∇
F
)
Φ
and
κ
∗
(G,
∇
G
)
Ψ
must
agree
modulo
p
i+1
.
Since
(F,
∇
F
)
Φ
is
indigenous
on
(Y
log
)
Φ
,
and
κ
∗
(G,
∇
G
)
Ψ
is
indigenous
on
the
s-pointed
stable
curve
of
genus
q
given
by
pulling
back
the
universal
one
by
Ψ
log
◦
κ
log
,
we
thus
obtain
that
the
diagram
in
the
Theorem
commutes
modulo
p
i+1
.
Then
since
(F,
∇
F
)
and
κ
∗
(G,
∇
G
)
agree
modulo
p
i
,
it
follows
that
their
underlying
vector
bundles
agree
modulo
p
i+1
.
By
a
similar
argument,
their
underlying
vector
bundles
also
agree
modulo
p
i+2
,
and
the
diagram
commutes
modulo
p
i+2
.
Then,
by
Lemma
2.5,
since
(F,
∇
F
)
∼
=
F
∗
(F,
∇
F
)
Φ
∗
∗
∗
Ψ
∗
∼
and
κ
(G,
∇
G
)
=
κ
F
(G,
∇
G
)
,
it
follows
that
(F,
∇
F
)
and
κ
(G,
∇
G
)
agree
modulo
p
i+1
.
This
completes
the
proof
of
the
induction
step.
§3.
Applications
of
the
Canonical
Frobenius
Lifting
In
this
Section,
we
apply
the
general
theory
of
§1
to
the
canonical
modular
Frobenius
lifting
constructed
in
§2.
In
particular,
we
define
the
notion
of
a
p-adic
quasiconformal
equivalence
class,
and
show
how
the
choice
of
such
a
class
allows
one
to
construct
both
affine
and
multiplicative
uniformizations
of
M
g,r
.
We
will
also
define
the
notion
of
a
p-
adic
topological
marking,
which
will
allow
us
to
construct
a
local
uniformization
of
M
g,r
by
means
of
the
affine
space
of
quadratic
differentials.
As
we
make
these
constructions,
we
will
compare
them
to
various
classical
constructions
in
the
complex
case.
Finally,
we
will
specialize
what
we
have
done
in
this
Chapter
to
the
case
of
elliptic
curves
(regarded
112
parabolically)
to
see
that
in
this
case,
the
canonical
Frobenius
lifting
corresponds
to
a
well-known
Frobenius
lifting
from
Serre-Tate
theory,
and
that,
consequently,
the
various
objects
constructed
from
it
–
i.e.,
canonical
curves,
modular
uniformizations,
etc.
–
reduce
to
the
corresponding
objects
of
classical
Serre-Tate
theory.
Canonical
Liftings
of
Curves
over
Witt
Vectors
ord
Let
N
g,r
;
Φ
log
N
be
as
in
the
last
subsection
of
§2.
Let
k
be
a
perfect
field
of
characteristic
p.
Let
A
=
W
(k),
the
ring
of
Witt
vectors
with
coefficients
in
k;
let
S
=
Spec(A).
Thus,
we
have
a
natural
Frobenius
automorphism
Φ
A
:
A
→
A
on
A.
Recall
the
notion
of
a
ord
canonical
liftings
of
A-valued
points
in
N
g,r
(Definition
1.9).
ord
Definition
3.1.
We
shall
call
a
point
α
0
∈
N
g,r
(k)
a
(p-adic)
quasiconformal
equivalence
class
(valued
in
k).
We
shall
call
an
r-pointed
stable
curve
of
genus
g
the
canonical
curve
in
the
class
α
0
if
it
admits
an
indigenous
bundle
such
that
the
pair
consisting
of
the
curve
ord
and
this
indigenous
bundle
defines
a
canonical
A-valued
point
of
N
g,r
whose
reduction
modulo
p
is
α
0
.
Remark.
Thus,
a
p-adic
quasiconformal
equivalence
class
consists
of
a
hyperbolically
ordi-
nary
r-pointed
stable
curve
(X
0
→
Spec(k);
p
1
,
.
.
.
,
p
r
:
Spec(k)
→
X
0
)
of
genus
g,
together
with
a
choice
of
a
nilpotent,
ordinary
indigenous
bundle
(E,
∇
E
)
0
on
X
0
log
.
Recall
from
Chapter
II,
Proposition
3.13,
that
for
a
given
ordinary
X
0
log
,
there
are
at
most
p
3g−3+r
possible
choices
for
(E,
∇
E
)
0
.
The
reason
for
attaching
the
term
“quasiconformal”
to
this
data
will
become
more
and
more
apparent
as
we
continue:
Namely,
unlike
the
complex
case
in
which,
once
g
and
r
are
determined,
all
curves
belong
to
the
same
quasiconformal
equiv-
alence
class,
the
uniformization
theory
that
we
shall
develop
in
this
paper
in
the
p-adic
setting
acts
(by
comparison
to
the
classical
complex
case)
as
if
there
are
many
different
quasiconformal
equivalence
classes
(for
a
given
g
and
r),
and
moreover,
this
equivalence
ord
class
is
determined
exactly
by
the
datum
of
a
point
in
N
g,r
.
Specializing
the
theory
of
§1,
we
obtain:
ord
Theorem
3.2.
For
every
p-adic
quasiconformal
equivalence
class
α
0
∈
N
g,r
(k),
there
ord
exists
a
canonical
lifting
α
∞
∈
N
g,r
(A),
i.e.,
more
concretely,
an
r-pointed
stable
curve
(X
→
Spec(A);
p
1
,
.
.
.
,
p
r
:
Spec(A)
→
X)
of
genus
g,
together
with
an
indigenous
(E,
∇
E
)
on
X
log
.
This
canonical
lifting
α
∞
is
uniquely
characterized
by
the
fact
that
it
is
fixed
log
log
under
Φ
−1
A
Φ
N
,
where
Φ
N
is
the
canonical
Frobenius
lifting
of
Theorem
2.8.
Corollary
3.3.
Suppose
that
the
pair
113
{(X
→
Spec(A);
p
1
,
.
.
.
,
p
r
:
Spec(A)
→
X);
(E,
∇
E
)}
is
canonical
(i.e.,
for
(E,
∇
E
),
this
means
that
it
is
the
restriction
of
the
(E
N
,
∇
E
N
)
of
Theorem
2.8).
Then
(1)
If
X
→
Spec(A)
is
smooth,
then
(E,
∇
E
)
defines
an
MF
∇
-object
on
X
log
in
the
sense
of
[Falt],
§2
(up
to
tensor
product
with
a
line
bundle
whose
square
is
trivial).
(2)
More
generally,
if
X
→
Spec(A)
is
not
smooth,
then
the
pair
is
obtained
by
gluing
together
(as
at
the
end
of
Chapter
I,
§2)
a
collection
of
smooth
canonical
pairs.
Proof.
The
two
statements
follow
by
specializing
Theorem
2.8.
Corollary
3.4.
A
pair
{(X
→
Spec(A);
p
1
,
.
.
.
,
p
r
:
Spec(A)
→
X);
(E,
∇
E
)}
consisting
of
a
smooth
r-pointed
curve
of
genus
g
and
an
indigenous
bundle
on
X
log
is
canonical
if
and
only
if
(1)
the
indigenous
bundle
(E,
∇
E
)
F
p
is
ordinary;
(2)
(E,
∇
E
)
defines
an
MF
∇
-object
on
X
log
in
the
sense
of
[Falt],
§2
(up
to
tensor
product
with
a
line
bundle
whose
square
is
trivial).
More
generally,
a
pair
consisting
of
an
r-pointed
stable
curve
of
genus
g
and
an
indigenous
bundle
on
X
log
is
canonical
if
and
only
if
it
is
obtained
by
gluing
together
canonical
pairs
as
at
the
end
of
Chapter
I,
§2.
Proof.
Let
us
first
consider
the
smooth
case.
By
the
previous
Corollary,
it
suffices
to
prove
the
“if”
part.
Since
(E,
∇
E
)
is
an
MF
∇
-object,
we
know
(by
Chapter
II,
Proposition
2.10)
that
(E,
∇
E
)
F
p
is
nilpotent.
Thus,
there
exists
a
canonical
pair
which
is
equal
to
our
given
pair
modulo
p.
On
the
other
hand,
it
follows
by
the
analogues
of
Lemmas
2.5
and
2.6
for
F
∗
over
S
=
Spec(A)
that
there
is
only
one
lifting
of
our
pair
modulo
p
that
admits
an
indigenous
bundle
which
is
an
MF
∇
-object.
Thus,
our
pair
must
be
the
canonical
pair.
Next,
we
consider
the
stable
case.
Again,
by
the
previous
Corollary,
it
suffices
to
prove
the
“if”
part.
As
before,
we
note
that
there
exists
a
canonical
pair
which
is
equal
to
our
given
pair
modulo
p.
By
the
previous
Corollary,
the
canonical
pair
is
obtained
by
gluing
together
smooth
canonical
pairs.
Since
a
smooth
canonical
lifting
is
unique,
it
thus
follows
that
our
pair
must
be
the
canonical
pair.
114
Corollary
3.5.
Suppose
that
we
have
an
r-pointed
(respectively,
s-pointed)
stable
curve
X
log
(respectively,
Y
log
)
of
genus
g
(respectively,
q)
over
S
log
(for
some
appropriate
choice
of
log
structure
on
S
log
).
Let
(E,
∇
E
)
(respectively,
(F,
∇
F
))
be
an
indigenous
bundle
on
X
log
(respectively,
Y
log
).
Suppose
that
(F,
∇
F
)
F
p
is
ordinary.
Let
φ
log
:
Y
log
→
X
log
be
log
admissible,
and
suppose
that
(F,
∇
F
)
=
φ
∗
(E,
∇
E
).
Then
the
pair
{X
log
;
(E,
∇
E
)}
is
canonical
if
and
only
if
the
pair
{Y
log
;
(F,
∇
F
)}
is
canonical.
ω
Proof.
First
note
that,
by
pulling
back
square
differentials
and
considering
Φ
ω
E
and
Φ
F
,
the
ordinariness
of
(F,
∇
F
)
F
p
implies
the
ordinariness
of
(E,
∇
E
)
F
p
.
The
stipulated
conditions
on
φ
log
imply
that
X
log
is
obtained
by
gluing
if
and
only
if
Y
log
is
obtained
by
gluing.
We
thus
reduce
to
the
smooth
case.
But
this
follows
immediately,
by
the
criterion
of
the
previous
Corollary
(about
the
indigenous
bundle
being
an
MF
∇
-object)
and
the
fact
that
F
∗
commutes
with
log
étale
coverings.
Canonical
Affine
Coordinates
on
M
g,r
We
maintain
the
notation
of
the
preceding
subsection.
Thus,
S
=
Spec(A);
A
=
W
(k);
ord
and
k
is
a
perfect
field
of
characteristic
p.
Let
α
∈
N
g,r
(A)
be
canonical.
Since
giving
ord
a
canonical
α
∈
N
g,r
(A)
is
equivalent
to
giving
the
p-adic
quasiconformal
equivalence
ord
class
α(mod
p)
∈
N
g,r
(k),
we
shall
frequently
abuse
notation
and
speak
of
“the
p-adic
quasiconformal
equivalence
class
α.”
Let
us
assume
that
α
corresponds
to
a
smooth
curve.
Then
applying
Theorems
1.12
and
1.14
to
the
canonical
Frobenius
lifting
of
Theorem
2.8
gives
the
following
results:
Theorem
3.6.
For
every
choice
of
a
p-adic
quasiconformal
equivalence
class
α,
we
obtain
a
local
canonical
uniformization
∼
α
U
can
:
Θ
aff
α
=
M
g,r
of
M
g,r
by
the
affine
space
modeled
on
Θ
α
.
Let
(B,
m
B
)
be
a
local
ring
with
residue
field
k
which
is
p-adically
complete
and
has
a
topologically
nilpotent
PD-structure
on
m
B
.
ord
Definition
3.7.
We
shall
say
that
β
∈
N
g,r
(B)
is
in
the
(p-adic)
quasiconformal
equiva-
ord
ord
lence
class
α
if
the
point
β(mod
m
B
)
∈
N
g,r
(k)
is
equal
to
α(mod
p)
∈
N
g,r
(k).
In
summary,
we
have
proven
the
following
result:
115
ord
Theorem
3.8.
For
every
β
∈
N
g,r
(B)
in
the
quasiconformal
equivalence
class
α,
we
obtain
a
canonical
class
κ
β
∈
m
B
·
Θ
β
,
as
well
as
a
local
uniformization
∼
β
U
β
:
Θ
aff
β
=
M
g,r
of
M
g,r
by
the
affine
space
modeled
on
Θ
β
.
Moreover,
this
uniformization
is
related
to
the
canonical
uniformization
by
tensoring
over
A
with
B,
applying
the
isomorphism
−1
:
(Θ
α
)
B
→
Θ
β
(Ψ
∨
αβ
)
ord
and
then
translating
by
κ
β
.
Finally,
for
all
β
∈
N
g,r
(B)
that
lie
in
the
quasiconformal
equivalence
class
α,
the
correspondence
β
→
κ
β
is
a
bijection
of
such
β
onto
m
B
·
Θ
β
.
Remark.
In
considering
the
uniformizations
just
obtained,
it
is
tempting
to
compare
them
with
the
local
uniformization
by
the
affine
space
modeled
on
the
tangent
space
to
M
g,r
given
in
the
complex
case
by
considering
geodesics
for
the
Teichmüller
metric.
We
believe,
however,
that
if
there
is
any
proper
complex
analogue
to
the
uniformizations
of
Theorems
3.6
and
3.8
at
all,
then
it
is
the
uniformization
obtained
by
Bers
coordinates.
Indeed,
unlike
the
Teichmüller
coordinates,
which
are
real,
but
not
complex
analytic,
the
Bers
coordinates
are
(complex)
analytic,
just
as
the
affine
coordinates
of
Theorems
3.6
and
3.8.
Also,
(perhaps
more
crucially)
the
Teichmüller
coordinates
are
the
same
for
elliptic
curves
regarded
either
hyperbolically
or
parabolically.
We
shall
soon
see,
however,
that
the
uniformizations
analogous
to
those
of
Theorems
3.6
and
3.8
for
elliptic
curves
treated
parabolically
are
different
from
those
in
the
hyperbolic
case.
One
difference
between
the
Bers
uniformization
and
the
uniformizations
of
Theorems
3.6
and
3.8
is
that
the
Bers
uniformization
is
by
the
affine
space
of
quadratic
differentials
(of
the
complex
conjugate
curve),
not
by
the
tangent
space
to
M
g,r
.
On
the
other
hand,
(even
in
the
complex
case)
one
cannot
have
a
holomorphic
local
canonical
uniformization
by
the
affine
space
of
quadratic
differentials,
as
one
can
see
easily
by
considering
a
one-pointed
curve
of
genus
one
with
an
automorphism
of
degree
three.
Thus,
to
obtain
a
uniformization
by
quadratic
differentials,
we
need
more
“rigidifying”
information.
In
our
case,
the
information
will
take
the
form
of
a
topological
marking
of
the
curve.
Topological
Markings
and
Uniformization
by
Quadratic
Differentials
We
maintain
the
notation
of
the
previous
subsection.
Let
us
consider
the
canonical
ord
ord
Frobenius
lifting
Φ
N
:
N
g,r
→
N
g,r
of
Theorem
2.8.
Just
as
in
Definition
1.2,
Φ
N
defines
ord
canonical
étale
local
systems
on
N
g,r
in
free
Z
p
-modules
of
rank
3g
−
3
+
r:
ord
Definition
3.9.
We
shall
refer
to
Θ
et
N
as
the
tangential
local
system
on
N
g,r
.
We
shall
ord
call
its
dual,
Ω
et
N
,
the
differential
local
system
on
N
g,r
.
116
Note
that
if
one
tensors
over
Z
p
with
F
p
,
then
these
local
systems
become
the
local
systems
(with
the
same
names)
considered
in
Chapter
II,
§3,
following
Proposition
3.4.
ord
Now
let
us
assume
that
k
is
algebraically
closed.
Let
α
∈
N
g,r
(A)
be
a
p-adic
quasi-
conformal
equivalence
class.
In
this
subsection,
however,
we
assume
that
α
corresponds
to
ord
a
smooth
curve.
We
would
like
to
consider
the
fundamental
groupoid
of
N
g,r
in
the
sense
of
[SGA
1],
Exposé
V,
p.
130.
Recall
that
this
fundamental
groupoid
is
the
category
of
ord
fiber
functors
from
the
category
of
finite
étale
coverings
of
N
g,r
to
the
category
of
sets.
ord
Moreover,
if
x
∈
N
g,r
(k),
then
x
defines
a
fiber
functor
F
x
of
this
type
by
simply
restrict-
ord
ord
ing
étale
coverings
of
N
g,r
to
Spec(k)
via
pull-back
by
x.
Thus,
if
x,
y
∈
N
g,r
(k),
then
we
shall
call
a
path
from
x
to
y
a
natural
transformation
from
F
x
to
F
y
.
On
the
other
hand,
recall
from
the
last
subsection
of
Chapter
II,
§3,
the
set
D
g,r
of
isomorphism
classes
of
totally
degenerate
r-pointed
stable
curves
of
genus
g.
If
Δ
∈
D
g,r
,
ord
then,
by
abuse
of
notation,
we
shall
also
write
Δ
∈
N
g,r
(A)
for
the
totally
degenerate
curve
over
A
represented
by
Δ.
Now
we
make
the
following
important
Definition
3.10.
We
shall
call
a
pair
μ
=
(Δ;
)
a
(p-adic)
topological
marking
for
the
ord
quasiconformal
equivalence
class
α
if
Δ
∈
D
g,r
and
is
a
path
from
α(mod
p)
∈
N
g,r
(k)
ord
to
Δ
∈
N
g,r
(k).
Let
μ
=
(Δ;
)
be
a
topological
marking
for
α.
Then
let
us
note
that
μ
defines
a
canonical
nondegenerate
bilinear
form
B
μ
on
Θ
α
as
follows.
First
note
that
by
the
construction
in
Proposition
3.6
of
Chapter
II,
§3
(which
is
carried
out
there
over
F
p
,
but
et
clearly
works
just
as
well
over
Z
p
),
we
have
a
canonical
nondegenerate
bilinear
form
B
Δ
et
∼
et
on
Θ
et
Δ
.
Now
the
path
defines
an
isomorphism
Θ
:
Θ
α
=
Θ
Δ
.
Thus,
if
we
pull-back
et
by
means
of
Θ
,
we
get
a
canonical
nondegenerate
bilinear
form
B
μ
et
on
Θ
et
B
Δ
α
.
Since
Θ
α
=
(Θ
et
)
⊗
A,
we
thus
obtain
(by
tensoring)
a
canonical
nondegenerate
bilinear
form
Z
p
α
B
μ
on
Θ
α
.
Now
let
(B,
m
B
)
be
a
local
ring
with
residue
field
k
which
is
p-adically
complete
and
has
ord
a
topologically
nilpotent
PD-structure
on
m
B
.
Let
β
∈
N
g,r
(B)
be
in
the
quasiconformal
equivalence
class
α.
Recall
the
canonical
isomorphism
−1
:
(Θ
α
)
B
→
Θ
β
(Ψ
∨
αβ
)
implicit
in
Theorem
3.8.
This
isomorphism
allows
us
to
transport
B
μ
to
Θ
β
so
as
to
obtain
a
canonical
nondegenerate
bilinear
form
B
μ
β
on
Θ
β
.
We
summarize
this
as
follows:
Proposition
3.11.
The
choice
of
a
topological
marking
μ
on
a
quasiconformal
equivalence
class
α
allows
one
to
define
a
canonical
nondegenerate
bilinear
form
B
μ
β
on
Θ
β
for
every
ord
β
∈
N
g,r
(B)
in
the
quasiconformal
equivalence
class
α.
117
This
finally
allows
us
to
give
local
uniformizations
of
M
g,r
by
means
of
quadratic
differen-
tials:
Namely,
we
compose
the
affine
uniformization
of
Theorem
3.8
with
the
isomorphism
Θ
β
∼
=
Ω
β
given
by
the
nondegenerate
bilinear
form
B
μ
β
:
Theorem
3.12.
The
choice
of
a
topological
marking
μ
on
a
quasiconformal
equivalence
class
α
that
corresponds
to
a
smooth
curve
allows
one
to
define
a
canonical
class
κ
∨
μ,β
∈
m
B
·
Ω
β
,
as
well
as
a
local
uniformization
∼
β
V
μ,β
:
Ω
aff
β
=
M
g,r
ord
of
M
g,r
by
the
affine
space
modeled
on
Ω
β
,
for
every
β
∈
N
g,r
(B)
in
the
quasiconformal
ord
equivalence
class
α.
Finally,
for
all
β
∈
N
g,r
(B)
that
lie
in
the
quasiconformal
equivalence
class
α,
the
correspondence
β
→
κ
μ,β
is
a
bijection
of
such
β
onto
m
B
·
Ω
β
.
Remark.
Thus,
we
have
obtained
a
canonical
uniformization
of
M
g,r
by
quadratic
differ-
entials
for
every
choice
of
a
topological
marking
on
α.
In
the
complex
case,
a
topological
marking
of
a
Riemann
surface
is
given
by
fixing
the
underlying
topological
manifold,
up
to
homeomorphisms
homotopic
to
the
identity.
Thus,
the
analogy
between
topological
markings
in
the
p-adic
and
complex
cases
lies
in
the
fact
that
a
p-adic
topological
marking
gives
one
a
canonical
basis
for
Θ
et
β
,
hence
for
Θ
β
,
corresponding
to
a
collection
of
parti-
tion
curves
(see
Introduction,
§2)
of
a
Riemann
surface.
This
specification
of
partition
curves
determines
a
topological
marking,
by
gluing
together
“pants”
along
the
partition
curves.
Thus,
instead
of
uniformizing
by
the
affine
space
modeled
on
Ω
β
,
we
could
also
have
uniformized
by
the
affine
space
modeled
on
a
direct
product
of
affine
lines,
one
for
each
“partition
curve.”
Whichever
choice
of
coordinates
(i.e.,
quadratic
differentials
or
partition
curves)
is
more
useful
depends
on
one’s
tastes
or
the
applications
one
has
in
mind.
Canonical
Multiplicative
Parameters
So
far
we
have
only
been
working
with
smooth
curves.
In
order
to
find
canonical
parameters
at
singular
curves,
we
need
to
work
with
multiplicative
parameters
(like
the
q-parameter
in
the
case
of
elliptic
curves),
as
opposed
to
affine
parameters,
as
in
Theorems
3.6
and
3.8.
ord
Let
α
∈
N
g,r
(A)
be
a
p-adic
quasiconformal
equivalence
class
(corresponding
to
a
curve
which
is
not
necessarily
smooth).
Let
us
assume,
for
the
rest
of
this
subsection,
that
k
is
algebraically
closed.
Let
Ω
log
α
be
the
restriction
of
Ω
M
log
to
α,
and
let
Θ
α
be
g,r
the
dual
A-module
to
Ω
log
α
.
Then
the
Frobenius
invariant
subsections
of
Θ
α
form
a
free
et
log
Z
p
-submodule
Θ
et
⊆
Θ
α
of
rank
3g
−
3
+
r.
Similarly,
we
have
Ω
α
⊆
Ω
α
.
Let
(M
g,r
)
α
be
α
the
completion
of
M
g,r
⊗
Z
p
A
at
the
image
of
α.
Let
ω
∈
Ω
et
α
have
residues
equal
to
zero
118
or
one
at
all
the
irreducible
components
of
the
divisor
at
infinity
of
M
g,r
,
and
nonzero
reduction
modulo
p.
Then,
just
as
in
Definition
1.11,
we
have
a
parameter
q
ω,α
on
(M
g,r
)
α
,
which
is
well-defined
up
to
multiplication
by
a
Teichmüller
representative
[k
×
].
This
parameter
is
a
unit
at
all
the
divisors
where
the
residue
of
ω
is
zero
and
has
valuation
one
at
all
the
divisors
where
the
residue
of
ω
is
one.
Moreover,
p
Φ
−1
N
(q
ω,z
)
=
q
ω,z
Definition
3.13.
We
shall
call
such
a
parameter
q
ω,α
a
canonical
multiplicative
parameter
on
(M
g,r
)
α
.
The
Case
of
Elliptic
Curves
Just
as
in
previous
Chapters,
it
is
useful
to
look
at
the
case
of
elliptic
curves
(regarded
parabolically)
since
the
calculations
are
usually
much
easier
in
this
case.
As
before,
we
let
log
log
M
1,0
be
the
log
stack
of
elliptic
curves,
and
f
log
:
G
log
→
M
1,0
be
the
universal
elliptic
curve
(with
the
log
structure
defined
by
the
pull-back
to
G
of
the
divisor
at
infinity
of
ord
M
1,0
).
Let
M
1,0
⊆
M
1,0
be
the
open
p-adic
formal
substack
parametrizing
ordinary
ord
ord
elliptic
curves.
Recall
that
we
computed
in
Chapter
II,
Theorem
3.11,
that
N
1,0
=
M
1,0
,
ord
and
that
the
section
of
S
1,0
over
(M
1,0
)
F
p
corresponding
to
the
unique
nilpotent,
ordinary
indigenous
bundle
on
an
elliptic
curve
was
given
explicitly
in
Example
2
of
Chapter
I,
§2.
Now
it
is
easy
to
see
that,
although
nominally
everything
in
this
Chapter
was
done
for
hyperbolic
curves,
much
of
the
theory
goes
through
for
elliptic
curves,
as
well.
In
ord
particular,
the
construction
of
the
canonical
Frobenius
lifting
Φ
N
on
M
1,0
goes
through
just
as
before.
Since
everything
else
in
the
Chapter
is
essentially
a
formal
consequence
of
the
existence
of
Φ
N
,
in
this
subsection,
we
would
like
to
compute
the
lifting
Φ
N
explicitly
for
elliptic
curves,
and
identify
the
resulting
concepts
(i.e.,
canonical
curves,
uniformization,
topological
marking,
etc.)
with
the
well-known
objects
of
classical
Serre-Tate
theory.
For
a
treatment
of
classical
Serre-Tate
theory,
we
refer
to
[Mess]
and
[KM]
(p.
260).
ord
Let
us
begin
by
recalling
a
certain
Frobenius
lifting
Φ
M
on
M
1,0
which
is
fundamental
to
Serre-Tate
theory.
Ultimately,
we
shall
show
that
Φ
M
=
Φ
N
.
First
recall
that
the
étale
quotient
of
the
(log)
p-divisible
group
P
associated
to
the
universal
elliptic
curve
ord
ord
G
ord
→
M
1,0
defines
a
local
system
L
on
M
1,0
in
free
Z
p
-modules
of
rank
one.
Also,
since
P
is
self-dual,
taking
Cartier
duals
gives
us
an
inclusion
L
∨
(1)
⊗
(Q
p
/Z
p
)
→
P
(where
the
“1”
in
parentheses
denotes
a
Tate
twist).
Let
P
Φ
⊆
P
be
the
subgroup
scheme
given
by
L
∨
(1)
⊗
(
p
1
Z
p
/Z
p
).
Thus,
taking
the
quotient
by
this
subgroup
scheme
P
Φ
gives
us
an
isogeny:
119
Φ
G
:
G
ord
→
H
ord
ord
ord
to
some
elliptic
curve
H
over
M
1,0
.
Let
Φ
M
:
M
1,0
→
M
1,0
be
the
classifying
morphism
of
H.
Thus,
H
=
Φ
∗M
G
ord
.
Since
considered
modulo
p,
the
subgroup
scheme
P
Φ
is
nothing
but
the
kernel
of
Frobenius,
it
follows
that
Φ
M
is
a
Frobenius
lifting,
and
that
(Φ
G
)
F
p
is
ord
just
the
relative
Frobenius
on
G
F
.
For
convenience,
we
shall
denote
objects
pulled
back
p
via
Φ
M
with
a
superscript
“F
.”
Now
let
us
consider
the
effect
of
pulling
back
the
indigenous
bundle
(E,
∇
E
)
F
on
(G
)
,
where
(E,
∇
E
)
is
the
indigenous
bundle
on
G
ord
given
in
Example
2
of
Chapter
I,
§2.
Let
(F,
∇
F
)
=
Φ
∗G
(E,
∇
E
)
F
.
Let
us
denote
by
ω
the
relative
dualizing
sheaf
of
ord
F
ord
G
ord
→
M
1,0
.
Then
as
a
vector
bundle,
F
=
Φ
∗G
(ω)
F
⊕
O
G
∗
F
Now
let
Φ
ω
G
:
Φ
G
(ω)
→
ω
denote
the
morphism
on
differentials
induced
by
Φ
G
,
divided
by
p.
Then
I
claim
that
Φ
ω
G
is
an
isomorphism.
Indeed,
since
we
are
dealing
with
ordinary
elliptic
curves,
the
local
group
structure
near
the
origin
is
isomorphic
to
that
of
G
m
(the
multiplicative
group
scheme),
and
the
Frobenius
lifting
Φ
G
just
amounts
to
the
p
th
power
map
on
G
m
.
This
proves
the
claim.
Since
E
=
ω
⊕
O
G
,
Φ
ω
G
thus
gives
us
an
isomorphism:
E∼
=
F
by
taking
the
direct
sum
of
Φ
ω
G
with
the
identity
on
O
G
.
Next,
we
consider
connections.
Recall
that
∇
E
differs
from
the
trivial
connection
by
the
tautological
Ad(E)-valued
dif-
ferential
form
given
by
mapping
the
first
factor
ω
to
the
second
factor
O
G
⊗
ω.
Thus,
when
we
pull-back
by
Φ
G
,
we
get
a
similar
nilpotent
endomorphism-valued
differential
form,
this
time
given
by
the
map
from
Φ
∗G
(ω)
F
(the
first
factor)
to
O
G
⊗
ω
(the
second
factor)
given
by
p
·
Φ
ω
G
.
On
the
other
hand,
when
we
compute
the
renormalized
Frobenius
pull-back
of
(E,
∇
E
),
we
divide
out
by
this
factor
of
p.
It
thus
follows
that
under
the
isomor-
phism
E
∼
=
F
considered
above,
the
renormalized
Frobenius
pull-back
gives
a
connection
on
F
which
corresponds
precisely
to
the
connection
∇
E
on
E.
Since
Φ
N
and
(E
N
,
∇
E
N
)
are
uniquely
characterized
by
the
property
that
the
renormalized
Frobenius
pull-back
of
Φ
∗N
(E
N
,
∇
E
N
)
is
isomorphic
to
(E
N
,
∇
E
N
),
we
thus
see
that
we
have
proven
the
following
result.
Theorem
3.14.
The
canonical
Frobenius
lifting
Φ
N
for
elliptic
curves
(regarded
parabol-
ically)
is
equal
to
the
Frobenius
lifting
Φ
M
.
Moreover,
the
canonical
indigenous
bundle
(E
N
,
∇
E
N
)
is
the
indigenous
bundle
constructed
in
Example
2
in
Chapter
I,
§2.
Remark.
In
other
words,
what
we
have
constructed
here
is
just
a
relative
version
of
the
uniformizing
MF
∇
-object
of
Definition
1.3.
120
Now
let
k
be
a
perfect
field
of
characteristic
p.
Then
it
is
well-known
from
Serre-Tate
theory
that
an
elliptic
curve
E
→
Spec(W
(k))
is
canonical
in
the
sense
of
Serre-Tate
the-
ory
if
and
only
if
the
point
in
α
∈
M
1,0
(W
(k))
that
it
defines
is
fixed
by
Φ
M
.
We
thus
obtain
that
the
definition
of
a
canonical
curve
given
in
Definition
3.1
is
consistent
with
the
definition
arising
from
Serre-Tate
theory.
Suppose
we
fix
a
trivialization
of
L
⊗2
|
α
.
Then
Serre-Tate
theory
gives
a
local
uniformization
of
M
1,0
near
this
point
α
by
the
comple-
tion
G
m
of
the
multiplicative
group
at
the
identity.
Relative
to
this
uniformization,
Φ
M
becomes
the
p
th
power
map
on
G
m
.
It
thus
follows
immediately
that
the
canonical
affine
parameters
that
we
constructed
before
(in
the
general
case)
correspond
to
the
logarithm
of
the
Serre-Tate
parameter
(up
to
multiplication
by
a
unit
of
W
(k)).
Moreover,
one
sees
⊗2
.
easily
that
the
local
system
Ω
et
M
corresponding
to
the
Frobenius
lifting
Φ
M
is
simply
L
Thus,
a
topological
marking
(in
the
sense
of
Definition
3.10)
defines
a
trivialization
of
L
⊗2
|
α
,
and
so
the
Serre-Tate
parameter
itself
is
a
canonical
multiplicative
parameter
in
the
sense
of
Definition
3.13.
We
summarize
this
as
follows:
Theorem
3.15.
Canonical
liftings
for
elliptic
curves
(as
defined
in
Definition
3.1
relative
to
Φ
N
)
are
the
same
as
canonical
liftings
in
the
sense
of
Serre-Tate
theory.
Moreover,
the
uniformization
of
Theorem
3.12
in
the
case
of
elliptic
curves
(regarded
parabolically)
is
the
same
as
the
uniformization
of
M
ord
1,0
given
by
Serre-Tate
theory.
Remark.
It
appears
that
the
case
discussed
here
in
Theorem
3.15,
i.e.,
the
case
g
=
1,
r
=
0,
is
the
only
case
of
the
theory
of
this
paper
that
is
essentially
a
reformulation
of
a
classically
known
theory.
For
instance,
already
in
the
case
g
=
1,
r
=
1,
despite
the
fact
that
ord
ord
and
N
1,1
are
quite
different.
M
1,0
=
M
1,1
(as
stacks),
it
is
not
difficult
to
show
that
N
1,0
ord
Indeed,
in
general,
there
exist
connected
components
of
N
1,1
that
are
of
degree
>
1
over
M
1,1
(cf.
Proposition
3.12
of
Chapter
II).
This
implies,
in
particular,
that
Φ
N
in
the
case
g
=
1,
r
=
1
is
quite
different
from
Φ
N
in
the
case
g
=
1,
r
=
0.
Finally,
we
observe
that
the
term
“topological
marking”
is
apt
in
this
case
in
the
sense
that
a
topological
marking
defines
a
trivialization
of
L|
α
,
which
is
analogous
in
the
complex
case
to
specifying
a
particular
pair
of
generators
for
the
fundamental
group
of
an
elliptic
curve.
121
Chapter
IV:
Canonical
Curves
§0.
Introduction
Because
canonical
curves
(as
defined
in
Chapter
III,
Definition
3.1)
admit
Frobenius
invariant
indigenous
bundles,
they
possess
a
number
of
special
arithmetic
and
geometric
properties.
In
this
Chapter,
we
study
a
number
of
these
properties,
foremost
among
which
are
the
existence
of
a
canonical
Frobenius
lifting,
and
the
construction
of
a
canonical
p-
divisible
group.
In
particular,
the
canonical
Frobenius
lifting
allows
us
to
give
a
geometric
characterization
of
canonical
curves
which
may
be
regarded
as
the
hyperbolic
analogue
of
the
statement
in
Serre-Tate
theory
that
a
lifting
of
an
ordinary
elliptic
curve
is
canonical
if
and
only
if
it
admits
a
lifting
of
Frobenius.
From
the
point
of
view
of
comparison
with
the
complex
case,
this
canonical
Frobenius
lifting
may
be
regarded
as
a
sort
of
p-adic
Green’s
function.
In
the
complex
case,
the
Green’s
function
plays
a
central
role
in
the
development
of
uniformization
theory
from
the
classical
(as
opposed
to
Bers’
quasiconformal)
point
of
view.
In
this
context,
the
Green’s
function
is
essentially
the
logarithm
of
the
hyperbolic
distance
function
between
two
points.
We
shall
see
that
the
Frobenius
lifting
also
gives
us
a
sort
of
p-adic
notion
of
distance.
Also,
we
shall
see
that
we
can
construct
“pseudo-Hecke
correspondences”
which
in
some
sense
geometrically
codify
this
notion
of
distance.
On
the
other
hand,
in
the
canonical
case,
we
can
also
construct
a
certain
Galois
rep-
resentation
(arising
from
the
torsion
points
of
the
canonical
log
p-divisible
group)
which
is
the
p-adic
analogue
of
the
canonical
representation
in
the
complex
case
of
the
fundamental
group
into
PSL
2
(R)
(defined
by
the
covering
transformations
of
the
upper
half
plane).
Thus,
in
some
sense,
we
see
that
at
least
in
the
canonical
case,
we
are
able
to
obtain
ana-
logues
of
most
of
the
fundamental
objects
that
appear
in
classical
complex
uniformization
theory.
This
brings
us
to
the
final
reason
for
wanting
to
study
the
canonical
case:
namely,
the
fact
that
the
universal
hyperbolically
ordinary
curve
(over
the
moduli
stack)
is
itself
(essentially)
a
canonical
curve.
Thus,
in
Chapter
V,
by
restricting
these
canonical
objects
over
the
universal
curve
to
a
given
(not
necessarily
canonical)
curve,
we
will
be
able
to
obtain
Green’s
functions,
canonical
Galois
representations,
and
so
on
for
noncanonical
curves,
as
well.
§1.
The
Canonical
Galois
Representation
In
this
Section,
we
construct
a
certain
canonical
Galois
representation
of
the
arith-
metic
fundamental
group
of
a
canonical
curve.
After
studying
some
of
the
basic
global
properties
of
such
representations,
we
then
consider
what
happens
on
the
ordinary
locus
of
the
curve.
In
particular,
we
construct
a
canonical
ordinary
Frobenius
lifting
over
the
ordinary
locus.
This
allows
us
to
apply
the
general
theory
of
Chapter
III,
§1.
We
will
122
refer
to
the
multiplicative
parameters
obtained
from
this
general
theory
as
the
Serre-Tate
parameters.
We
will
make
use
of
the
Serre-Tate
parameter
quite
often
in
this
Chapter.
Throughout
this
Section,
we
will
work
over
A
=
W
(k),
where
k
is
a
perfect
field
of
odd
characteristic.
The
quotient
field
of
A
will
be
denoted
by
K.
Let
g,
r
be
nonnegative
ord
integers
such
that
2g
−
2
+
r
≥
1.
Also,
we
will
deal
with
a
fixed
α
∈
N
g,r
(A),
which
corresponds
to
a
smooth
canonical
curve
f
log
:
X
log
→
S
log
,
where
S
log
is
Spec(A)
with
the
trivial
log
structure.
Since
singular
canonical
curves
are
just
obtained
by
gluing
together
smooth
canonical
curves,
we
shall
concentrate
mainly
on
the
smooth
case.
Construction
and
Global
Properties
Let
(E,
∇
E
)
be
the
canonical
indigenous
bundle
on
X
log
(whose
existence
is
stated
in
Chapter
III,
Theorem
2.8).
In
fact,
unless
the
number
r
of
marked
points
is
even,
such
a
vector
bundle
will
not
exist.
However,
one
can
always
pass
to
an
étale
double
cover
of
X
on
which
it
will
exist,
and
then
descend.
For
simplicity,
we
will
just
act
as
though
this
problem
does
not
exist,
except
when
we
state
final
results
in
Theorems,
in
which
case
our
representations
will
be
into
GL
±
(that
is,
the
general
linear
group
GL
modulo
the
subgroup
{±1}).
Now,
we
would
also
like
to
say
that
the
renormalized
Frobenius
pull-back
F
∗
(E,
∇
E
)
F
is
isomorphic
to
(E,
∇
E
).
In
general,
this
may
only
be
true
up
to
tensoring
with
a
line
bundle
with
connection
whose
square
is
trivial,
but
this
may
also
be
ignored,
provided
we
remember
that
ultimately
our
representations
will
be
into
GL
±
,
not
GL.
Let
us
choose
an
isomorphism
Φ
E
:
(E,
∇
E
)
∼
=
F
∗
(E,
∇
E
)
F
which
is
the
identity
on
determinants.
We
shall
call
Φ
E
the
canonical
Frobenius
action
on
(E,
∇
E
).
Now
let
us
assume
that
there
exists
a
rational
point
x
:
S
→
X
on
X
which
avoids
log
the
marked
points.
Let
us
denote
by
Π
the
profinite
group
π
1
(X
K
,
x
K
).
Then
Theorem
2.6
of
[Falt]
implies
that
Theorem
1.1.
There
exists
a
unique
dual
crystalline
(in
the
sense
of
[Falt],
§2)
repre-
sentation
ρ
:
Π
→
GL
±
(V
)
(where
V
is
a
free
Z
p
-module
of
rank
two)
that
corresponds
(under
the
functor
D(−)
of
[Falt],
§2)
to
(E,
∇
E
,
Φ
E
).
Moreover,
the
determinant
representation
of
ρ
is
the
cyclotomic
character.
We
shall
refer
to
ρ
as
the
canonical
crystalline
representation
associated
to
X
log
.
Remark.
In
the
complex
case,
a
hyperbolic
Riemann
surface
can
be
uniformized
by
the
upper
half
plane.
Then
the
fundamental
group
of
the
Riemann
surface
acts
on
the
upper
half
plane
via
covering
transformations,
and
so
we
get
a
representation
of
the
fundamental
123
group
into
PSL
2
(R),
which
is
canonically
determined
up
to
conjugation.
The
representa-
tion
ρ
of
Theorem
1.1
is
the
p-adic
analogue
of
this
complex
representation.
def
Now
let
Δ
=
π
1
(X
K
,
x
K
)
be
the
geometric
subgroup
of
Π,
so
Γ
=
Π/Δ
is
the
Galois
group
of
K
over
K.
Then
by
“the
comparison
theorem”
(Theorem
5.3
of
[Falt]),
we
get:
Theorem
1.2.
Let
p
≥
5.
Then
the
group
cohomology
modules
H
i
(Δ,
Ad(V
)(1))
(where
the
“(1)”
is
a
Tate
twist)
are
zero,
except
when
i
=
1.
Let
U
=
H
1
(Δ,
Ad(V
)(1)).
Then
U
is
a
crystalline
Z
p
-Γ-module,
which,
as
a
Z
p
-module
is
free
of
rank
6(g
−
1)
+
2r.
It
corresponds
under
the
functor
D(−)
to
an
MF-object
(in
the
sense
of
Fontaine-Laffaille)
(N
;
F
i
(N
);
φ
i
)
over
A,
where
N
is
a
free
A-module
of
rank
6(g
−
1)
+
2r;
F
i
(N
)
=
0
if
log
⊗2
i
≥
4;
F
i
(N
)
=
N
if
i
≤
0;
F
i
(N
)
is
naturally
isomorphic
to
H
0
(X,
(ω
X/S
)
(−D)),
if
1
1
i
=
1,
2,
3;
and
N/F
(N
)
is
naturally
isomorphic
to
H
(X,
τ
X
log
/S
log
).
Remark.
Some
mathematicians
have
raised
questions
concerning
that
the
validity
of
the
proof
in
[Falt],
Theorem
5.3.
However,
in
this
one-dimensional
case,
one
can
give
ad
hoc
proofs
of
this
result,
and,
moreover,
(at
least
in
the
closed
case,
when
r
=
0)
T.
Tsuji
has
orally
informed
the
author
that
he
has
obtained
a
different
proof
of
Theorem
5.3
of
[Falt].
One
interesting
fact
about
the
canonical
representation
ρ
is
that
it
is
possible
to
characterize
it
–
as
well
as
the
canonicality
of
X
log
–
solely
in
terms
of
the
properties
of
ρ
as
a
Galois
representation:
Theorem
1.3.
Suppose
that
p
≥
5.
Let
X
log
→
Spec(A)
be
any
(not
necessarily
canonical)
r-pointed
smooth
curve
of
genus
g
over
A.
Assume
that
we
are
not
in
the
cases
(g
=
0;
r
=
3)
or
(g
=
1;
r
=
1).
Let
τ
:
Π
→
GL
±
(W
)
be
any
dual
crystalline
representation
of
log
Π
=
π
1
(X
K
,
x
K
)
on
a
free
Z
p
-module
W
of
rank
two
such
that
(1)
H
P
i
(Δ,
Ad(W
)(1))
=
0
if
i
=
1;
H
1
(Δ,
Ad(W
)(1))
is
crystalline,
and
corresponds
to
an
MF-object
M
=
(M
;
F
i
(M
);
ψ
i
)
such
that
F
i
(M
)
=
0
if
i
≥
4;
F
i
(M
)
=
M
if
i
≤
0;
and
F
i
(M
)
is
a
free
A-module
of
rank
3(g
−
1)
+
r
if
i
=
1,
2,
3;
(2)
the
Frobenius
endomorphism
of
(M/F
1
(M
))
F
p
(arising
from
the
MF-
object
of
(1))
is
an
isomorphism;
(3)
det(τ
)
is
the
cyclotomic
character.
Then
X
log
is
canonical,
and
τ
is
isomorphic
to
the
representation
ρ
of
Theorem
1.1.
Proof.
Since
τ
is
asserted
to
be
dual
crystalline,
it
corresponds
to
some
vector
bundle
with
connection
(G,
∇
G
)
on
X
log
,
together
with
a
filtration
F
i
(G)
on
G.
Let
i
1
(respectively,
i
2
)
be
the
largest
i
such
that
F
i
(G)
=
0
(respectively,
F
i
(G)
=
G).
Thus,
i
1
≥
i
2
.
The
124
condition
that
det(τ
)
be
cyclotomic
implies
that
i
1
+
i
2
=
1.
If
the
rank
of
F
i
1
(G)
is
not
one,
then
i
1
=
i
2
,
and
det(τ
)
could
not
be
cyclotomic,
so
F
i
1
(G)
must
be
of
rank
one,
and
i
1
>
i
2
.
Let
L
=
F
i
1
(G).
Thus,
L
is
a
line
bundle.
Let
j
1
be
the
largest
j
such
that
F
j
(M
)
=
0.
Now
we
claim
that
L
can
not
be
stable
under
∇
G
.
Indeed,
if
it
were,
then
the
monodromy
at
the
marked
points
of
∇
G
on
L,
being
nilpotent
and
one-dimensional,
must
be
zero.
Thus,
the
induced
connection
on
L
has
no
poles
at
the
marked
points.
But
this
would
imply
that
deg(L)
=
0.
Hence
the
rank
over
A
of
F
j
1
(M
)
would
be
≤
h
0
(X,
L
⊗2
⊗
ω
X/S
)
≤
g
<
3g
−
3
+
r
(by
Clifford’s
Theorem),
which
contradicts
our
hypotheses.
This
proves
the
claim.
On
the
other
hand,
by
Griffiths
transversality,
if
i
1
−
i
2
≥
2,
then
F
i
1
(G)
would
have
to
be
stable
under
∇
G
.
Thus,
i
1
=
i
2
+
1,
so
i
1
+
i
2
=
1
implies
that
i
1
=
1
and
i
2
=
0.
Now
rank
A
(F
3
(M
))
=
3g
−
3
+
r
≤
h
0
(X,
L
⊗2
⊗
ω
X/S
),
so
the
line
bundle
L
⊗2
⊗
ω
X/S
must
be
nonspecial,
by
Clifford’s
Theorem.
It
thus
follows
that
deg(L
⊗2
)
≥
2g−2+r.
Since
the
Kodaira-Spencer
morphism
for
the
filtration
is
nonzero,
we
cannot
have
deg(L
⊗2
)
>
2g
−
2
+
r.
Thus,
we
see
that
(G,
∇
G
)
must
be
indigenous.
Since
it
is
also
carried
to
itself
by
the
renormalized
Frobenius,
it
follows
from
Chapter
III,
Corollary
3.4,
that
X
log
is
canonical,
and
that
(G,
∇
G
)
must
be
the
canonical
indigenous
bundle
of
Theorem
1.1.
Remark.
For
the
reader
who
is
interested
in
handling
the
cases
g
=
0;
r
=
3
and
g
=
1;
r
=
1,
as
well,
we
remark
that
by
considering
conditions
(similar
to
those
imposed
on
H
i
(Δ,
Ad(W
)(1)))
on
higher
symmetric
powers
of
W
,
one
can
characterize
the
canonical
representations
in
these
cases
as
well
solely
in
terms
of
their
properties
as
Galois
represen-
tations.
Remark.
Really,
the
substantive
missing
element
here
is
that
it
is
not
clear
to
the
author
how
to
characterize
the
property
of
being
“dual
crystalline”
solely
in
terms
of
proper-
ties
of
the
representation
relative
to
the
triple
(Π;
Δ
⊆
Π;
Π/Δ
∼
=
Gal(K/K)).
Thus,
log
is
always
present
in
the
background
of
this
ultimately,
a
knowledge
of
the
curve
X
“Galois
representation-theoretic”
characterization
of
the
canonical
representation.
For
in-
stance,
if
the
property
of
being
“dual
crystalline”
were
known
to
depend
only
on
the
triple
(Π;
Δ
⊆
Π;
Π/Δ
∼
=
Gal(K/K)),
then
one
could
obtain
the
result
that
whether
or
not
a
curve
is
canonical
depends
only
on
that
triple.
In
the
following,
we
return
to
the
assumption
that
X
log
is
canonical.
The
Horizontal
Section
over
the
Ordinary
Locus
We
maintain
the
notation
of
the
previous
subsection.
Let
X
ord
be
the
p-adic
formal
scheme
which
is
the
open
sub-formal
scheme
of
X
given
by
the
complement
of
the
super-
singular
divisor
(Chapter
II,
Proposition
2.6).
Let
us
endow
X
ord
with
the
log
structure
induced
by
X
log
,
and
call
the
resulting
log
formal
scheme
(X
log
)
ord
.
We
shall
refer
to
125
(X
log
)
ord
as
the
ordinary
locus
of
X
log
.
The
purpose
of
this
subsection
is
to
prove
and
interpret
the
following
result:
1
Theorem
1.4.
There
exists
a
unique
subbundle
T
2
⊆
E|
X
ord
of
rank
one
with
the
following
properties:
1
(1)
T
2
is
horizontal,
and
moreover,
for
any
n,
the
reduced
line
bundle
1
2
T
Z/p
n
Z
has
a
nonempty
subsheaf
(in
the
category
of
sets)
consisting
of
1
2
horizontal
sections
that
generate
T
Z/p
n
Z
as
an
O
X
ord
-module;
1
(2)
T
2
is
taken
to
itself
by
Φ
E
.
Finally,
(T
2
)
⊗2
is
naturally
isomorphic
to
τ
X
log
/S
log
|
X
ord
.
1
1
Proof.
Let
us
prove
that
there
exists
a
unique
Φ
E
-invariant
horizontal
subbundle
T
2
⊆
E|
X
ord
with
horizontal
generating
sections.
We
prove
this
by
induction
on
n,
where
the
n
th
1
step
is
the
construction
of
such
a
T
2
modulo
p
n
.
For
n
=
1,
recall
that
(up
to
tensoring
with
a
line
bundle)
E
F
p
is
an
FL-bundle
(Chapter
II,
Proposition
2.5).
Then
under
the
1
correspondence
of
that
Proposition,
we
take
our
subbundle
T
F
2
p
to
be
the
subbundle
of
E
corresponding
to
the
subbundle
that
we
called
“T
”
in
our
discussion
of
FL-bundles
in
Chapter
II,
§1.
This
subbundle
is
clearly
horizontal,
and
has
local
generating
sections
that
are
horizontal.
In
this
case,
uniqueness
follows
from
the
fact
that
the
p-curvature
is
nonzero.
Now
we
assume
that
n
≥
2,
and
that
the
result
is
known
for
n−1.
Let
U
log
=
(X
log
)
ord
,
and
let
Φ
log
:
U
log
→
(U
log
)
F
be
a
Frobenius
lifting.
Let
us
consider
the
quotient
Q
(respectively,
P)
of
E
Z/p
n
Z
by
p
n−1
·
F
1
(E)
(respectively,
p
n−1
·
E).
Thus,
P
is
a
quotient
of
Q,
and
P
=
E
Z/p
n−1
Z
.
Let
T
⊆
P
be
the
subbundle
given
us
by
the
induction
hypothesis.
Let
T
⊆
Q
be
the
the
inverse
image
of
T
⊆
P
via
the
surjection
Q
→
P.
1
∗
F
2
Then
Φ
∗
(T
)
F
⊆
Φ
∗
Q
F
defines
a
subbundle
T
Z/p
n
Z
of
F
(E)
Z/p
n
Z
.
It
follows
from
the
1
1
2
definition
of
T
F
2
p
and
the
fact
that
we
are
on
the
ordinary
locus
that
T
Z/p
n
Z
is
flat
over
1
2
Z/p
n
Z.
The
existence
of
local
horizontal
generating
sections
for
T
Z/p
n
Z
follows
by
taking
such
a
section
of
T
,
lifting
it
to
T
,
and
then
pulling
back
this
lifted
section
of
T
to
a
1
∗
2
section
of
T
Z/p
n
Z
via
Φ
.
That
the
connection
vanishes
on
this
section
follows
from
the
definitions,
plus
the
fact
that
pulling
back
by
Φ
adds
an
extra
factor
of
p.
Since
T
is
Φ
E
-
1
1
n−1
2
2
Z
=
T
.
Thus,
by
the
construction
of
T
Z/p
invariant,
it
follows
that
T
Z/p
n
Z
⊗
Z/p
n
Z
,
1
2
it
is
clear
that
T
Z/p
by
Φ
E
will
n
Z
is
Φ
E
-invariant,
since
pulling
back
any
lifting
of
T
1
2
give
T
Z/p
n
Z
.
Also,
this
same
observation
(coupled
with
the
induction
hypothesis)
proves
uniqueness.
This
completes
the
proof
of
the
induction
step.
The
last
statement
about
1
1
(T
2
)
⊗2
follows
from
considering
the
splitting
of
the
Hodge
filtration
that
T
2
defines.
126
Now
suppose
that
our
basepoint
x
:
S
→
X
maps
into
the
ordinary
locus
X
ord
.
Let
Π
ord
=
π
1
((X
log
)
ord
K
,
x
K
).
Thus,
we
have
a
natural
morphism:
Π
ord
→
Π
Let
us
denote
the
restriction
of
ρ
to
Π
ord
via
this
natural
morphism
by
ρ
ord
:
Π
ord
→
GL
±
(V
ord
)
Then
if
we
apply
the
theory
of
[Falt],
§2
to
interpret
Theorem
1.4,
we
see
that
the
subbundle
1
T
2
⊆
E|
X
ord
in
fact
defines
a
sub-MF
∇
-object
corresponding
to
an
étale
representation
(ρ
ord
)
et
:
Π
ord
→
GL
±
(V
et
)
of
Π
ord
,
for
some
rank
one
free
Z
p
-quotient
module
V
ord
→
V
et
.
Here,
by
“étale,”
we
mean
that
the
kernel
of
(ρ
ord
)
et
defines
an
étale
covering
of
X
ord
.
In
other
words,
we
have
an
exact
sequence
of
(“up
to
{±1}”)
representations
of
Π
ord
:
0
→
V
et
∨
(1)
→
V
ord
→
V
et
→
0
where
the
“1”
in
parentheses
is
a
Tate
twist.
We
state
this
as
a
Corollary:
Corollary
1.5.
The
restriction
ρ
ord
of
ρ
to
Π
ord
defines
an
(“up
to
{±1}”)
module
V
ord
of
Π
ord
,
which
fits
into
an
exact
sequence:
0
→
V
et
∨
(1)
→
V
ord
→
V
et
→
0
where
V
et
is
étale
and
of
rank
one
over
Z
p
.
The
Canonical
Frobenius
Lifting
over
the
Ordinary
Locus
In
this
subsection,
we
construct
the
generalized
analogue
(for
an
arbitrary
canonical
X
)
of
the
p-adic
endomorphism
of
the
ordinary
locus
of
the
moduli
stack
of
elliptic
curves
obtained
by
sending
an
elliptic
curve
with
ordinary
reduction
to
its
quotient
modulo
its
unique
subgroup
scheme
which
is
étale
locally
isomorphic
to
μ
p
.
In
many
respects,
the
construction
is
similar
to
(although
not
literally
a
logical
consequence
of)
the
construction
ord
of
the
Frobenius
lifting
on
N
g,r
constructed
in
Chapter
III,
§2.
log
Consider
the
canonical
indigenous
bundle
(E,
∇
E
)
(of
Theorem
1.1)
on
the
canonical
curve
X
log
→
S
log
.
By
Chapter
II,
Proposition
2.5,
(E,
∇
E
)
F
p
corresponds
to
an
FL-bundle
0
→
T
F
p
→
F
→
O
X
F
p
→
0
127
on
X
F
log
.
By
the
material
directly
preceding
Chapter
II,
Proposition
1.2,
splittings
of
this
p
log
exact
sequence
correspond
to
Frobenius
liftings
on
X
Z/p
2
Z
.
Now,
over
the
ordinary
locus
log
of
X
,
the
Hodge
filtration
defines
such
a
splitting.
Let
us
denote
the
resulting
Frobenius
lifting
on
the
ordinary
locus
by
log
ord
F
Φ
2
:
(X
log
)
ord
)
)
Z/p
2
Z
Z/p
2
Z
→
((X
Let
us
denote
by
E
the
vector
bundle
which
is
the
inductive
limit
of
the
following
diagram:
F
1
(E)
⏐
⏐
p·
−→
E
F
1
(E)
where
the
horizontal
arrow
is
the
natural
inclusion.
Note
that
E
Z/p
n
Z
depends
only
on
F
E
Z/p
n
Z
.
By
the
definition
of
the
renormalized
Frobenius
pull-back,
Φ
∗
2
E
Z/p
2
Z
is
naturally
isomorphic
to
E
Z/p
2
Z
.
We
shall
identify
these
two
sheaves
in
the
following
discussion.
On
the
other
hand,
by
considering
the
object
in
the
upper
right-hand
corner
of
the
diagram
we
obtain
a
morphism
defining
E,
F
Φ
∗
2
E
Z/p
2
Z
→
E
Z/p
2
Z
∗
1
F
∗
1
F
whose
restriction
to
Φ
∗
2
F
1
(E)
F
Z/p
2
Z
vanishes
on
p
·
Φ
2
F
(E)
Z/p
2
Z
and
maps
Φ
2
F
(E)
Z/p
2
Z
into
p
·
F
1
(E)
Z/p
2
Z
(by
the
definition
of
the
correspondence
between
Frobenius
liftings
and
splittings
of
the
FL-bundle
F).
log
ord
F
Now
let
Ψ
3
:
(X
log
)
ord
)
)
Z/p
3
Z
be
any
lifting
of
Φ
2
.
Then,
again
from
Z/p
3
Z
→
((X
the
definition
of
the
renormalized
Frobenius
pull-back,
we
obtain
a
morphism
F
Ψ
∗
3
E
Z/p
3
Z
→
E
Z/p
3
Z
which
vanishes
on
p
2
·
Ψ
∗
3
F
1
(E)
F
Z/p
3
Z
.
However,
if
Ψ
3
is
an
arbitrary
lifting
of
Φ
2
,
then
we
∗
1
F
don’t
know
that
Ψ
3
F
(E)
Z/p
3
Z
is
mapped
into
F
1
(E)
Z/p
3
Z
.
Now
suppose
that
we
modify
Ψ
3
by
a
section
δ
∈
Γ(X
ord
,
T
F
p
).
Let
H
T
:
T
F
p
→
(τ
X
log
/S
log
)
F
p
|
X
ord
be
the
isomorphism
defined
by
projecting
to
the
Hodge
filtration.
Then
the
subsheaf
of
E
Z/p
3
Z
given
by
the
image
of
F
1
(E)
F
Z/p
3
Z
under
Ψ
3
+
δ
differs
from
the
corresponding
image
subsheaf
under
Ψ
3
by
the
amount
H
T
(δ)
∈
Γ(X
ord
,
(τ
X
log
/S
log
)
F
p
).
Indeed,
this
follows
from
the
definitions,
plus
the
fact
that
the
Kodaira-Spencer
morphism
for
E
is
the
identity.
Since
H
T
is
an
isomorphism,
it
thus
follows
that
there
exists
a
unique
Frobenius
lifting
128
log
ord
F
Φ
3
:
(X
log
)
ord
)
)
Z/p
3
Z
Z/p
3
Z
→
((X
1
that
lifts
Φ
2
such
that
Φ
∗
3
maps
F
1
(E)
F
Z/p
3
Z
into
F
(E).
Clearly,
we
may
repeat
this
procedure
modulo
p
n
for
arbitrary
n
≥
3,
so
as
to
obtain
a
unique
log
ord
)
→
((X
log
)
ord
)
F
Φ
log
X
:
(X
such
that
under
the
natural
morphism
Φ
∗
X
E
F
→
E
the
Hodge
filtration
is
preserved.
Note,
moreover,
that
it
follows
from
the
fact
that
the
Kodaira-Spencer
morphism
at
the
Hodge
section
is
an
isomorphism
plus
the
interpretation
of
the
FL-bundle
F
in
terms
of
Frobenius
liftings
that
this
Frobenius
lifting
Φ
log
X
is
ordinary
in
the
sense
of
Chapter
III,
Definition
1.1.
In
summary,
we
have
proven
the
following
result:
Theorem
1.6.
Let
X
log
be
a
canonical
curve;
(E,
∇
E
)
the
canonical
indigenous
bundle
on
X
log
.
Then
there
exists
a
unique
ordinary
Frobenius
lifting
(called
canonical)
log
ord
Φ
log
)
→
((X
log
)
ord
)
F
X
:
(X
over
the
ordinary
locus
that
preserves
the
Hodge
filtration.
In
particular,
we
can
apply
the
theory
of
Chapter
III,
§1,
to
the
Frobenius
lift-
∇
ing
Φ
log
X
.
Note
that
it
follows
immediately
from
the
definitions
that
the
MF
-object
(E,
F
1
(E),
∇
E
,
Φ
E
)|
(X
log
)
ord
is
precisely
the
uniformizing
MF
∇
-object
associated
to
Φ
log
X
(as
in
Chapter
III,
Definition
1.3).
Let
us
write
T
=
(T
2
)
⊗2
1
Thus,
∇
E
(respectively,
Φ
E
)
induces
a
natural
connection
(respectively,
Frobenius
action)
on
T
,
which
defines
the
canonical
tangential
local
system
of
Chapter
III,
Definition
1.2.
Since
T
is
a
line
bundle,
it
is
the
same
to
give
(over
an
étale
covering
of
X
ord
)
a
generating
Frobenius
invariant
section
of
it,
or
a
generating
Frobenius
invariant
section
of
its
dual.
Thus,
(just
as
in
Chapter
III,
Definition
1.11)
if
θ
is
such
a
section
of
T
,
then
θ
defines,
at
every
z
∈
X
ord
(A)
that
avoids
the
marked
points,
a
unique
multiplicative
parameter
q
θ
∈
R
z
×
129
(where
R
z
is
the
completion
of
X
ord
at
z).
If
the
residue
of
θ
is
equal
to
one
at
a
marked
point
z
∈
X
ord
(A),
then
we
get
a
multiplicative
parameter
q
∈
R
z
(with
valuation
one
at
the
divisor
Im(z))
which
is
unique
up
to
multiplication
by
a
Te-
ichmüller
representative
[k
×
].
Definition
1.7.
We
shall
call
q
(respectively,
q
θ
)
the
Serre-Tate
parameter
(respectively,
relative
to
θ)
at
z.
Note,
in
particular,
that
by
the
theory
of
Chapter
III,
§1,
Φ
−1
X
maps
q
(respectively,
q
θ
)
to
p
p
q
(respectively,
q
θ
).
Remark.
In
some
sense,
it
would
be
more
aesthetically
pleasing
if
one
could
obtain
the
Frobenius
lifting
of
Theorem
1.6
in
the
following
way.
We
consider
the
universal
curve
ord
C
→
N
g,r
.
Then
C
parametrizes
(r
+
1)-pointed
stable
curves
of
genus
g,
so
we
have
a
Frobenius
lifting
on
some
stack
which
is
étale
over
C.
If
we
could
prove
that
this
Frobenius
ord
lifting
is
compatible
with
the
canonical
Frobenius
lifting
on
N
g,r
,
then
we
could
obtain
a
canonical
Frobenius
lifting
on
C
(or
at
least
some
stack
étale
over
C)
simply
by
using
the
ord
canonical
Frobenius
on
N
g,r+1
.
The
problem
with
this
approach
is
that
despite
the
fact
that
the
canonical
modular
Frobenius
liftings
of
Chapter
III
do
have
many
interesting
functorial
relations
(i.e.,
rela-
tive
to
restriction
to
the
boundary
and
log
admissible
coverings),
in
general,
the
sort
of
compatibility
of
Frobenius
liftings
necessary
to
make
the
above
sketch
of
a
proof
work
–
namely,
compatibility
with
“forgetting
a
marked
point”
–
simply
does
not
hold.
Indeed,
one
can
already
see
this
in
the
case
of
the
morphism
M
1,1
→
M
1,0
which
is
the
identity
on
the
underlying
stacks,
but
which
we
think
of
as
assigning
to
a
one-
pointed
curve
of
genus
one
the
underlying
elliptic
curve.
Here,
the
canonical
Frobenius
on
ord
ord
N
1,1
cannot
be
compatible
with
the
canonical
Frobenius
on
N
1,0
for
the
following
reason.
ord
Since
N
1,0
→
M
1,0
is
an
open
immersion,
it
would
follow
that
the
canonical
Frobenius
ord
on
N
1,1
would
descend
to
an
open
formal
subscheme
of
M
1,1
.
But
this
would
mean
that
even
if
a
one-pointed
curve
of
genus
one
in
characteristic
p
belongs
to
several
distinct
quasiconformal
equivalence
classes
(a
phenomenon
which
by
Chapter
II,
Proposition
3.13,
does
occur),
the
canonical
liftings
of
that
curve
would
be
the
same
for
all
quasiconformal
equivalence
classes.
But
this
would
mean
that
we
have
several
different
ordinary
indigenous
bundles
on
a
single
hyperbolic
curve,
all
of
which
are
Frobenius
invariant.
By
Chapter
III,
Lemma
2.6,
this
is
absurd.
130
§2.
The
Canonical
Log
p-divisible
Group
Although
the
existence
of
the
canonical
Galois
representation
of
§1
is,
in
and
of
itself,
of
some
interest,
one
technical
drawback
that
it
has
is
that
it
is
difficult
to
relate
the
log
that
it
properties
of
the
Galois
representation
or
the
characteristic
zero
coverings
of
X
Q
p
log
determines
to
X
Z/p
n
Z
.
Thus,
in
this
Section,
we
shall
construct
a
log
p-divisible
group
log
on
X
which
gives
us
back
the
canonical
Galois
representation
(by
looking
at
the
Galois
action
on
torsion
points),
but
which
has
the
advantage
that
one
can
study
and
understand
its
reductions
modulo
p
n
in
a
similar
fashion
to
the
elliptic
modular
case
(which
is
studied
in
[KM]).
Log
p-divisible
Groups
at
Infinity
We
maintain
the
notation
of
the
previous
Section
(although
k
need
not
be
algebraically
closed,
just
perfect).
For
basic
facts
about
log
schemes,
we
refer
to
[Kato]
and
[Kato2].
In
[Kato2],
certain
finite,
log
flat
group
objects
over
the
compactified
moduli
stack
of
elliptic
curves
are
introduced
which
are
supposed
to
be
the
analogue
at
infinity
of
the
usual
finite,
flat
group
schemes
that
one
gets
from
elliptic
curves
by
considering
the
kernel
of
multiplication
by
a
power
of
p.
Since
we
will
use
such
objects
(as
well
as
the
p-divisible
group
objects
obtained
by
taking
direct
limits
thereof)
later
in
this
Section,
we
take
the
time
out
in
the
present
subsection
to
review
explicitly
the
construction
of
these
finite,
log
flat
group
objects.
Let
R
=
A[[q]]
be
a
complete
local
ring
which
is
formally
smooth
of
dimension
one
over
A.
If
one
inverts
q,
then
by
taking
the
(p
n
)
th
root
of
q,
one
obtains
an
extension
of
finite
flat
group
schemes
0
→
Z/p
n
Z(1)
→
G
n
→
Z/p
n
Z
→
0
over
R[1/q].
Because
q
is
not
a
unit
in
R,
it
is
impossible
to
extend
this
extension
of
finite
flat
group
schemes
over
R[1/q]
to
an
extension
of
finite
flat
group
schemes
over
R.
Our
goal
in
this
subsection,
however,
is
to
exhibit
a
natural
extension
of
the
above
exact
sequence
to
an
exact
sequence
defined
over
R
by
working
with
group
objects
in
the
category
of
finite,
log
flat
log
schemes
over
Spec(R)
log
.
(In
this
subsection,
we
will
regard
Spec(R)
as
endowed
with
the
log
structure
arising
from
the
divisor
defined
by
q.)
For
nonnegative
integers
a,
b,
let
M
a,b
be
the
monoid
given
by
taking
the
quotient
of
N
(where
N
is
the
monoid
of
nonnegative
integers)
by
the
equivalence
relation
generated
by
(p
a
,
0)
∼
(0,
b).
Let
e
1
∈
M
a,b
(respectively,
e
2
∈
M
a,b
)
be
the
image
of
(1,
0)
(respec-
tively,
(0,
1))
in
M
a,b
.
Then
it
follows
from
the
theory
of
[Kato2]
(especially,
§4.1,
5.1)
that
we
can
construct
the
desired
extension
2
log
0
→
Z/p
n
Z(1)
→
G
n
→
Z/p
n
Z
→
0
131
n
as
follows:
For
j
∈
{0,
.
.
.
,
p
n
−
1},
consider
the
scheme
G
n,j
given
by
R[x]/(x
p
−
q
j
),
with
the
log
structure
given
by
the
chart
([Kato],
§2)
M
n,j
with
e
1
→
x;
e
2
→
q.
Denote
log
the
resulting
log
scheme
by
(G
)
log
n,j
.
Let
G
n,j
be
the
universal
valuative
log
space
([Kato2],
log
§1.3.1)
(which,
in
this
case,
will
still
be
a
log
scheme)
associated
to
(G
)
log
n,j
.
Let
G
n
be
log
log
the
union
of
the
G
log
n,j
.
Note
that
when
we
invert
q,
G
n
becomes
G
n
.
Endow
G
n
with
log
the
unique
structure
of
group
object
that
extends
the
group
structure
on
G
n
.
Then
G
n
is
a
group
object
in
the
category
of
finite,
log
flat
log
schemes
over
Spec(R)
log
,
and
it
fits
into
an
exact
sequence
as
above.
log
log
As
we
allow
n
to
vary,
we
get
morphisms
G
n
→
G
n+1
.
Thus,
we
obtain
an
ind-group
object
G
over
Spec(R)
log
.
Definition
2.1.
We
shall
refer
to
G
as
the
log
p-divisible
group
over
Spec(R)
log
obtained
by
taking
p
th
power
roots
of
q
∈
R.
Finally,
we
remark
that,
although
what
we
are
doing
here
is,
in
some
sense,
just
“trivial
general
nonsense,”
its
utility
lies
in
the
fact
that
by
using
it,
we
can
obtain
p-adic
finite
coverings
of
X
log
that
are
defined
over
all
of
X
log
,
thus
allowing
us
to
algebrize.
Construction
of
the
Canonical
Log
p-divisible
Group
We
now
turn
to
the
construction
of
the
canonical
log
p-divisible
group
on
X
log
.
Con-
sider
the
MF
∇
-object
(E,
∇
E
,
Φ
E
),
defined
by
the
canonical
indigenous
bundle.
Let
n
≥
1.
Let
U
⊆
X
be
the
open
p-adic
subscheme
defined
by
removing
the
marked
points.
Then
the
reduction
modulo
p
n
of
(E,
∇
E
,
Φ
E
)|
U
defines,
by
[Falt],
Theorem
7.1,
a
finite,
flat
group
scheme
(annihilated
by
p
n
),
which
we
denote
by
G
n
|
U
→
U
.
On
the
other
hand,
let
R
be
the
complete
local
ring
at
any
one
of
the
marked
points.
Then
(E,
∇
E
,
Φ
E
)
defines
a
Serre-Tate
parameter
(as
in
Definition
1.7)
q
∈
R/[k
×
].
Let
q
∈
R
be
any
representative
of
q.
Then
R
=
A[[
q
]].
Let
G
log
n
|
R
be
the
log
scheme
constructed
in
the
previous
sub-
n
th
section
by
taking
a
(p
)
-root
of
q.
Observe
that
different
choices
of
q
give
us
naturally
isomorphic
G
log
,
then
G
log
n
|
R
’s.
Also,
note
that
if
we
invert
q
n
|
R
becomes
(G
n
|
U
)|
R
.
Thus,
log
we
see
that
G
n
|
U
and
the
various
G
n
|
R
at
the
marked
points
glue
together
naturally
to
log
form
a
finite,
log
flat
group
object
G
log
,
which
a
priori
is
just
p-adic,
but
may
be
n
→
X
algebrized
since
X
is
proper
over
A.
Also,
as
n
varies,
we
obtain
natural
morphisms
log
.
.
.
→
G
log
n
→
G
n+1
→
.
.
.
which
thus
form
an
inductive
system
of
group
objects.
Definition
2.2.
We
shall
call
this
inductive
system
of
group
objects
the
canonical
log
p-divisible
group
on
X
log
.
132
Remark.
As
usual,
strictly
speaking
we
really
have
only
defined
a
“group
up
to
{±1}.”
That
is,
we
really
only
have
a
group
object
over
(perhaps)
a
finite
étale
covering
of
X
of
degree
4,
plus
descent
data
(satisfying
the
cocycle
condition
up
to
{±1})
down
to
the
original
X.
We
could,
of
course,
develop
the
general
nonsense
of
such
“groups
up
to
{±1},”
but
we
choose
not
to,
since
it
seems
to
serve
no
real
purpose.
If
we
invert
p,
then
this
log
p-divisible
group
G
log
on
X
log
defines
a
local
system
on
log
)
in
free
Z
p
-modules
of
rank
two.
Thus,
we
get
a
Galois
representation
the
étale
site
(X
Q
p
et
on
the
Tate
module
T
of
characteristic
zero
torsion
points
of
G
log
:
ρ
G
log
:
Π
→
GL
±
(T
)
Then
we
have
the
following
result
(which
is
immediate
from
the
theory
of
[Falt],
especially
the
construction
in
the
proof
of
Theorem
7.1):
Proposition
2.3.
The
representation
ρ
G
log
is
isomorphic
to
the
canonical
Galois
repre-
sentation
ρ
of
Theorem
1.1.
Review
of
the
Theory
of
[Katz-Mazur]
In
this
subsection,
we
apply
to
the
log
p-divisible
group
G
log
the
theory
of
[KM],
which
is
exposed
in
[KM]
solely
in
the
case
of
the
canonical
log
p-divisible
on
the
compact-
ified
moduli
stack
of
elliptic
curves,
but
whose
proofs
go
through
without
change
for
the
canonical
log
p-divisible
group
G
log
on
any
canonical
curve
X
log
.
First
of
all,
because
G
log
is
a
logarithmic
p-divisible
group,
it
follows
from
[Mess],
Chapter
II,
Theorem
3.3.13,
that
if
we
consider
the
formal
neighborhood
of
the
identity
section
:
X
→
G,
we
obtain
a
formally
smooth
formal
scheme
G(
)
over
X,
which
is
easily
seen
to
have
relative
dimension
1
over
X.
We
would
like
to
use
this
observation
to
apply
the
theory
of
[KM],
Chapter
1,
on
“A-generators”
and
“A-structures”
to
G
log
.
The
theory
there
goes
through
just
as
in
the
modular
case
since
the
only
technical
assumption
needed
on
the
finite,
flat
(logarithmic)
group
schemes
whose
A-generators
we
wish
to
parametrize
is
that
they
be
closed
subschemes
of
some
smooth
one-dimensional
scheme.
However,
looking
at
the
proofs
of
[KM],
one
sees
that
in
fact,
it
suffices
to
have
the
finite,
flat
(log)
group
schemes
be
closed
subschemes
of
a
formally
smooth
formal
scheme
(such
as
G(
))
of
relative
dimension
one.
Thus,
we
can
define
various
moduli
problems,
just
as
in
[KM],
Chapter
3,
by
means
of
various
structures:
(1)
a
Γ(n)-structure,
which
consists
of
giving
a
Drinfeld
basis
for
G
log
n
;
(2)
a
Γ
1
(n)-structure,
which
consists
of
giving
a
point
“of
exact
order
p
n
”
in
G
log
n
;
133
(3)
a
Γ
0
(n)-structure,
which
consists
of
giving
an
isogeny
G
log
→
H
log
(where
H
log
is
also
a
log
p-divisible
group)
whose
kernel
is
cyclic
of
order
p
n
.
Moreover,
just
as
in
[KM],
one
proves
that
these
various
moduli
problems
are
representable
by
schemes
X(n);
X
1
(n);
X
0
(n)
that
are
finite
over
X.
Finally,
all
of
these
schemes
X(n);
X
1
(n);
and
X
0
(n)
are,
in
fact,
regular.
Indeed,
away
from
the
marked
points,
the
proofs
of
regularity
in
[KM],
Chapters
5
and
6,
boil
down
to
general
nonsense
plus
two
technical
results
(Proposition
5.3.4
and
Theorem
6.1.1).
Since
these
technical
results
are
proven,
respectively,
for
arbitrary
formal
groups
and
arbitrary
finite
group
schemes,
it
is
immediate
that
the
regularity
proofs
of
[KM]
in
the
modular
case
go
through
without
change
for
X(n);
X
1
(n);
and
X
0
(n).
At
the
marked
points,
the
combinatorial
descriptions
of
the
situation
at
the
cusps
in
[KM],
Chapter
10,
go
through
without
change
for
the
above
moduli
problems.
We
thus
obtain
the
following
Theorem:
Theorem
2.4.
The
schemes
X(n);
X
1
(n);
and
X
0
(n)
that
represent
the
moduli
problems
listed
above
are
all
regular,
and
hence
equal
to
the
normalizations
of
X
in
the
finite
cover-
ings
of
X
K
defined
by
the
appropriate
composites
of
ρ
:
Π
→
GL
±
(V
)
with
finite
quotients
GL
±
(V
)
→
G,
just
as
in
the
classical
modular
case.
Fix
a
positive
integer
n.
We
shall
also
need
to
analyze
X
0
(n)
modulo
p,
in
a
fashion
similar
to
what
is
done
in
the
modular
case
in
[KM],
Chapter
13.
Let
us
(for
the
rest
of
the
Section)
denote
X
0
(n)
by
Y
,
and
let
us
use
a
subscript
m
on
X,
Y
,
etc.,
to
denote
reduction
modulo
p
m+1
.
Let
us
denote
by
Φ
A
:
A
→
A,
Φ
k
:
k
→
k
the
respective
absolute
Frobenius
morphisms,
and
by
a
superscript
F
m
the
result
of
base-changing
an
object
by
the
m
th
power
of
Frobenius,
and
by
Φ
X
0
:
X
0
→
X
0
F
the
relative
Frobenius
of
X
0
.
Essentially,
the
description
of
Y
0
=
Y
⊗
Z
p
F
p
given
in
[KM],
Chapter
13,
goes
through
in
our
situation
here,
but
we
need
to
do
things
with
a
little
bit
more
care,
since
[KM]
often
falls
back
on
the
“crutch”
of
using
the
modular
interpretation
of
their
“X,”
which
we
lack
in
this
more
general
situation.
For
each
ordered
pair
of
nonnegative
integers
(a,
b)
such
that
a
+
b
=
n,
we
would
like
to
define
a
k-scheme
X
0
(a,
b)
of
“(a,
b)-cyclic
isogenies”
together
with
a
k-morphism
ι
(a,b);0
:
X
0
(a,
b)
→
Y
0
.
We
do
this
as
follows.
If
a,
b
≥
1,
then
we
let
X
0
(a,
b)
be
the
schematic
inverse
image
of
Inf
p−1
(Δ)
(the
(p
−
1)
st
infinitesimal
neighborhood
of
the
diagonal)
via
b−1
F
Φ
a−1
X
0
×
Φ
X
0
:
X
0
×
X
0
a−b
→
X
0
F
a−1
×
X
0
F
a−1
If
a
or
b
is
zero,
then
we
let
X
0
(a,
b)
be
the
schematic
inverse
image
of
the
diagonal
Δ
via
Φ
aX
0
×
Φ
bX
0
:
X
0
×
X
0
F
134
a−b
a
→
X
0
F
×
X
0
F
a
Observe
that
in
either
case,
(X
0
(a,
b))
red
is
smooth
over
k;
Φ
aX
0
×
Φ
bX
0
maps
X
0
(a,
b)
⊆
a−b
a
a
into
Δ
⊆
X
0
F
×
X
0
F
;
and
X
0
(a,
b)
comes
equipped
with
a
finite,
flat,
radicial
X
0
×
X
0
F
morphism
D
0
(a,
b)
:
X
0
(a,
b)
→
X
0
To
define
ι
(a,b);0
:
X
0
(a,
b)
→
Y
0
we
must
specify
a
cyclic
subgroup
of
order
p
n
of
D
0
(a,
b)
∗
G
log
0
.
Now
on
the
one
hand,
by
composing
the
a
th
power
of
Frobenius
with
the
b
th
-power
of
the
Verschiebung
(as
in
[KM],
Theorem
13.3.5),
we
get
some
subgroup
object
of
order
p
n
of
D
0
(a,
b)
∗
G
log
0
,
and
by
the
same
argument
as
that
given
in
[KM],
Theorem
13.3.5,
one
sees
that
this
subgroup
must
be
cyclic
(in
the
Drinfeldian
sense).
Thus,
by
the
modular
definition
of
Y
0
,
we
get
a
morphism
ι
(a,b);0
:
X
0
(a,
b)
→
Y
0
.
In
order
to
apply
the
theory
of
[KM],
Chapter
13,
we
must
verify
the
conditions
(1)
through
(8)
listed
at
the
beginning
of
that
Chapter.
(Caution:
The
letters
X
and
Y
in
[KM],
Chapter
13,
are
used
in
the
reverse
way
to
the
way
that
they
are
used
here.)
Conditions
(1),
(2),
(4),
(5),
and
(6)
are
trivial.
Condition
(3)
follows
from
the
regularity
of
Y
and
the
fact
that
over
a
supersingular
point,
there
is
only
one
A-generator
valued
in
k
for
a
cyclic
group,
namely
the
identity
element.
Note
that
at
ordinary
points,
one
can
do
the
same
analysis
of
p
th
power
isogenies
of
log
p-divisible
groups
as
is
done
in
[KM],
Chapter
13,
§3.
Thus,
Condition
(7)
(that
ι
(a,b);0
is
a
closed
immersion)
and
Condition
(8)
(that
the
ι
(a,b);0
’s
define
an
isomorphism
of
the
disjoint
union
of
X
0
(a,
b)’s
with
Y
0
over
the
ordinary
locus)
follow
at
the
level
of
topological
spaces
from
this
analysis,
and
at
the
level
of
complete
local
rings
by
considering
the
deformation
parameters
for
the
domain
and
range
log
p-divisible
groups
of
the
isogeny.
We
thus
get
a
result
analogous
to
[KM],
Theorem
13.4.7:
Theorem
2.5.
The
k-scheme
Y
0
is
the
disjoint
union,
with
crossings
at
the
supersingular
points
(in
the
terminology
of
[KM],
Chapter
13,
§1),
of
the
n
+
1
schemes
X
0
(a,
b)
(where
a
+
b
=
n).
Let
f
(a,b)
∈
k[[x,
y]]
be
the
equation
a−1
(x
p
a
b−1
−
y
p
)
p−1
b
if
a,
b
≥
1,
and
let
it
be
x
p
−
y
p
if
a
or
b
is
zero.
Then
the
completed
local
ring
at
a
k-rational
supersingular
point
of
Y
0
is
isomorphic
to
k[[x,
y]]/(
(a,b)
135
f
(a,b)
)
with
the
closed
subscheme
X
0
(a,
b)
⊆
Y
0
given
by
the
equation
f
(a,b)
.
§3.
The
Compactified
Canonical
Frobenius
Lifting
In
this
Section,
we
study
the
canonical
Frobenius
lifting
on
the
ordinary
locus
of
a
canonical
curve
(defined
in
Theorem
1.6).
In
particular,
we
study
its
behavior
at
su-
persingular
points,
and
“compactify
it”
in
some
sense,
so
as
to
obtain
“pseudo-Hecke
correspondences.”
It
is
by
abstracting
the
main
properties
of
this
compactified
Frobenius
in
the
canonical
case
that
we
shall
obtain
a
geometric
criterion
for
a
curve
to
be
canonical
in
§4.
The
Canonical
Frobenius
Lifting
and
the
Canonical
Log
p-divisible
Group
Let
us
denote
by
log
ord
Φ
log
)
→
((X
log
)
ord
)
F
X
:
(X
the
canonical
Frobenius
lifting
of
Theorem
1.6.
Let
G
log
be
the
canonical
log
p-divisible
group
on
X
log
of
Definition
2.2.
Then
we
rephrase
Theorem
1.6
in
terms
of
G
log
as
follows:
Theorem
3.1.
The
canonical
Frobenius
lifting
of
Theorem
1.6
log
ord
Φ
log
)
→
((X
log
)
ord
)
F
X
:
(X
induces
an
isogeny
of
degree
p
Φ
∗
X
(G
log
)
F
|
(X
log
)
ord
→
G
log
|
(X
log
)
ord
between
the
canonical
log
p-divisible
groups
that
lifts
the
Frobenius
morphism
modulo
p.
log
ord
)
that
has
this
property.
Moreover,
Φ
log
X
is
the
unique
Frobenius
lifting
over
(X
Proof.
The
existence
of
the
isogeny
follows
from
the
fact
that
we
have
defined
a
morphism
between
the
respective
Dieudonné
crystals
that
respects
the
Hodge
filtrations.
This
induces
the
isogeny
(see
[BBM]
and
[Mess]).
On
the
other
hand,
the
uniqueness
statement
follows
from
the
uniqueness
statement
in
Theorem
1.6,
together
with
the
fact
that
if
a
Frobenius
lifting
induces
such
an
isogeny,
it
automatically
preserves
the
Hodge
filtrations
on
the
Dieudonné
crystals.
136
Let
n
≥
0.
Let
Y
=
X
0
(n).
Let
Y
ord
⊆
Y
be
the
p-adic
open
formal
subscheme
consisting
of
points
lying
over
X
ord
.
Now
the
Frobenius
lifting
of
Theorem
3.1
allows
us
to
extend
the
decomposition
in
characteristic
p
of
Y
F
p
into
components
corresponding
to
(a,
b)-cyclic
isogenies
to
a
decomposition
over
A
=
W
(k),
on
the
ordinary
locus.
To
obtain
this
decomposition,
we
define
closed
p-adic
subschemes
X(a,
b)
ord
⊆
X
ord
×
A
(X
ord
)
F
a−b
via
the
same
recipe
as
we
did
for
X
0
(a,
b),
except
using
our
canonical
Frobenius
lifting
log
ord
Φ
log
)
→
((X
log
)
ord
)
F
instead
of
Φ
X
0
.
Then,
just
as
before,
we
get
a
natural
X
:
(X
embedding
ι
(a,b)
:
X(a,
b)
ord
→
Y
ord
(analogous
to
ι
(a,b);0
)
which
induces
an
isomorphism
Y
ord
∼
=
X(a,
b)
ord
(disjoint
union)
a+b=n
Finally,
over
Y
ord
,
we
have
a
tautological
isogeny
G
log
→
H
Y
log
ord
Y
ord
(where
G
log
is
the
pull-back
of
G
log
to
Y
ord
)
such
that
over
X(a,
b)
ord
,
H
Y
log
ord
is
naturally
Y
ord
a−b
a−b
isomorphic
to
the
pull-back
of
(G
log
)
F
via
the
projection
X(a,
b)
ord
→
(X
ord
)
F
to
the
second
factor.
Local
Analysis
at
Supersingular
Points
We
now
exploit
the
existence
of
the
isogeny
of
Theorem
3.1
to
understand
the
be-
havior
of
the
canonical
Frobenius
lifting
at
the
supersingular
points.
Let
x
∈
X(k)
be
a
supersingular
point.
In
studying
x,
we
will
often
need
to
involve
its
various
Frobenius
i
conjugates
x
F
∈
X(k)
(which
may
be
infinite
in
number
if
the
perfect
field
k
is
not
finite).
We
begin
our
analysis
by
considering
the
double
iterate
of
the
Frobenius
morphism
over
some
infinitesimal
neighborhood
V
⊆
X
F
p
at
x:
Φ
2
V
:
G
log
|
V
→
G
log
|
V
F
2
Thus,
V
is
the
spectrum
of
a
local
artinian
ring,
with
residue
field
k.
Let
us
assume
that
V
is
contained
in
the
supersingular
divisor
(Chapter
II,
Proposition
2.6)
of
the
canonical
indigenous
bundle.
By
definition,
this
means
that
over
V
,
the
Hodge
filtration
coincides
with
the
FL-bundle
filtration.
It
thus
follows
that
over
V
,
the
kernels
of
the
Verschiebung
and
Frobenius
morphisms
coincide.
Since
the
kernel
of
the
composite
of
the
Verschiebung
and
the
Frobenius
is
just
the
kernel
of
multiplication
by
p,
it
follows
that
the
morphism
Φ
2
V
is
isomorphic
to
the
morphism
“multiplication
on
p.”
In
particular,
it
follows
that
137
G
log
|
V
F
2
∼
=
G
log
|
V
By
iterating
this
isomorphism,
we
obtain
that
G
log
|
V
is
isomorphic
to
the
pull-back
to
V
of
a
p-divisible
group
over
k.
Since
the
Kodaira-Spencer
morphism
of
the
Hodge
filtration
of
(E,
∇
E
)
is
an
isomorphism,
this
implies
that
V
must
be
Spec(k).
Thus,
the
assumption
that
V
lies
inside
the
supersingular
divisor
implies
that
V
is
reduced.
Put
another
way,
we
see
that
we
have
proven
(in
this
general
context)
the
analogue
of
Igusa’s
theorem
([KM],
p.
355):
Proposition
3.2.
The
supersingular
divisor
of
the
canonical
indigenous
bundle
(E,
∇
E
)
is
étale
over
k.
Next,
let
us
observe
that
for
any
x
∈
X(k),
the
completed
local
ring
R
x
of
X
at
x
(which
is
formally
smooth
of
dimension
one
over
A)
is
naturally
isomorphic
to
the
universal
deformation
space
of
the
p-divisible
group
G
log
|
x
.
Indeed,
it
follows
from
the
theory
of
[Mess]
that
deformations
of
G
log
|
x
are
given
by
deformations
of
the
Hodge
filtration;
thus,
our
observation
follows
from
the
fact
that
the
Kodaira-Spencer
morphism
of
the
Hodge
filtration
of
(E,
∇
E
)
is
an
isomorphism.
Now
suppose
that
x
∈
X(k)
is
supersingular.
Then
the
isomorphism
G
log
|
x
F
2
∼
=
G
log
|
x
obtained
above
from
the
double
iterate
of
Frobenius
induces
a
natural
isomorphism
of
complete
local
rings
Ψ
x
:
R
x
F
2
∼
=
R
x
which
will
play
an
important
role
in
the
sequel.
Now
fix
a
number
n
≥
1,
and
let
Y
=
X
0
(n).
If
x
∈
X(k),
let
us
denote
by
X
x
the
formal
spectrum
of
X
at
x,
i.e.,
Spf(R
x
).
We
will
use
similar
notation
for
Y
.
Over
Y
,
we
have
a
tautological
cyclic
isogeny
of
order
p
n
:
log
G
log
Y
→
H
Y
Fix
a
supersingular
x
∈
X(k).
By
the
analysis
of
[KM],
reviewed
in
§2,
there
exists
a
unique
y
∈
Y
(k)
lying
over
x.
Now
by
thinking
of
the
completed
local
rings
of
X
as
universal
classifying
spaces,
we
obtain
a
morphism:
(D
y
,
R
y
)
:
Y
y
→
X
x
×
A
X
x
F
n
138
where
D
y
is
the
classifying
morphism
for
the
“domain
p-divisible
group”
G
log
Y
,
and
R
y
is
log
the
classifying
morphism
for
the
“range
p-divisible
group”
H
Y
.
Here
we
use
the
fact
that
restricted
to
x,
the
tautological
isogeny
is
just
the
n
th
iterate
of
the
Frobenius
morphism,
so
H
Y
log
|
y
=
G
log
.
Also,
note
that
by
the
deformation
theory
of
[Mess],
a
deformation
of
the
x
F
n
tautological
isogeny
is
uniquely
determined
by
the
induced
deformations
of
the
domain
and
range
p-divisible
groups.
It
thus
follows
that
the
morphism
(D
y
,
R
y
)
is
formally
unramified,
hence
a
closed
immersion.
Thus,
henceforth,
we
shall
think
of
Y
y
as
a
formal
divisor
inside
X
x
×
X
x
F
n
by
means
of
the
closed
immersion
(D
y
,
R
y
).
One
of
the
most
important
properties
of
this
divisor
Y
y
is
its
symmetry.
More
precisely,
given
Y
y
,
one
can
obtain
a
divisor
in
X
x
F
n
×
X
x
in
two
ways:
(1)
by
applying
the
isomorphism
X
x
×
X
x
F
n
∼
=
X
x
F
n
×
X
x
given
by
switch-
ing
the
two
factors;
(2)
by
conjugating
first
by
the
n
th
power
of
Frobenius,
so
as
to
obtain
a
divisor
in
X
x
F
n
×
X
x
F
2n
and
then
applying
the
isomorphism
Spf(Ψ
x
)
−n
to
the
second
factor.
Then
we
claim
that
these
two
divisors
in
X
x
F
n
×
X
x
are
the
same.
Indeed,
to
see
this,
it
suffices
to
trace
what
happens
to
the
tautological
isogeny.
Let
us
consider
the
second
procedure
stated
above.
First,
we
conjugate
the
tautological
isogeny
by
Frobenius:
n
F
→
(H
Y
log
)
F
(G
log
Y
y
)
y
n
Since
at
y,
this
isogeny
is
just
the
n
th
iterate
of
the
Frobenius
morphism,
by
looking
at
Dieudonné
modules,
it
follows
that
the
kernel
of
this
isogeny
is
contained
in
the
kernel
of
multiplication
by
p
n
.
Thus,
we
get
a
morphism
n
F
(H
Y
log
)
F
→
(G
log
Y
y
)
y
n
n
2n
F
Since
(H
y
log
)
F
=
(G
log
,
we
thus
see
that
the
divisor
in
X
x
F
n
×
X
x
obtained
this
y
)
way
is
just
the
divisor
of
isogenies
(lifting
the
n
th
iterate
of
Frobenius)
from
the
universal
deformation
of
G
log
x
(pulled
back
from
the
second
factor)
to
the
universal
deformation
of
log
F
n
(G
x
)
(pulled
back
from
the
first
factor).
On
the
other
hand,
it
we
look
at
the
divisor
obtained
from
the
first
procedure
stated
above,
it
admits
exactly
the
same
description.
This
proves
the
claim.
The
next
important
property
of
this
divisor
Y
y
⊆
X
x
×
X
x
F
n
is
that
if
we
restrict
it
to
the
ordinary
locus,
it
becomes
equal
to
the
union
of
the
“local
versions”
of
the
schemes
X(a,
b)
ord
(described
at
the
end
of
the
preceding
subsection).
More
precisely,
X(a,
b)
ord
a−b
is
defined
as
a
closed
subscheme
of
X
ord
×
(X
ord
)
F
.
Thus,
we
obtain
X(a,
b)
ord
|
X
x
⊆
a−b
(X
ord
×
(X
ord
)
F
)
X
x
by
restricting
to
the
formal
scheme
X
x
.
Since
a
−
b
≡
a
+
b
=
139
n(mod
2),
by
applying
the
appropriate
power
of
Ψ
x
,
we
thus
obtain
a
subscheme,
which
we
shall
call
n
ord
X(a,
b)
ord
×
(X
ord
)
F
)
X
x
x
⊆
(X
Then
it
follows
from
the
functorial
definition
of
Y
(in
terms
of
parametrizing
isogenies)
that
Y
y
|
X
ord
=
X(a,
b)
ord
x
(disjoint
union)
a+b=n
Let
us
denote
the
supersingular
divisor
of
X
by
X
ss
.
We
are
now
ready
to
summarize
what
we
have
done:
Definition
3.3.
We
shall
call
the
pair
def
H
x
=
(Y
y
;
Ψ
x
)
consisting
of
the
divisor
Y
y
⊆
X
x
×
X
x
F
n
and
the
isomorphism
Ψ
x
the
n
th
canonical
local
Hecke
correspondence
of
X
log
at
x.
We
shall
call
the
triple
(Y
→
X;
Φ
log
X
;
{H
x
}
x∈X
ss
)
the
n
th
canonical
pseudo-Hecke
correspondence
of
X
log
.
Of
course
ideally,
the
local
Hecke
correspondences
H
x
would
glue
together
to
form
a
n
global
Hecke
correspondence,
i.e.,
a
morphism
Y
→
X
×
X
F
,
just
as
in
the
classical
case
with
modular
curves.
We
shall
investigate
this
issue
in
the
next
subsection,
but
(unfortu-
nately)
what
we
shall
find
is
that
the
existence
of
such
global
Hecke
correspondences
is
a
rather
rare
phenomenon.
Global
Hecke
Correspondences
We
maintain
the
notation
of
the
previous
subsection.
Definition
3.4.
We
shall
say
that
the
canonical
curve
X
log
admits
a
global
n
th
canonical
Hecke
correspondence
if
there
exists
a
morphism
(D,
R)
:
Y
→
X
×
X
F
140
n
that
is
equal
to
the
local
morphisms
(D
y
,
R
y
)
of
the
preceding
subsection
for
every
y
∈
Y
lying
over
a
supersingular
point
of
X.
We
say
that
X
log
is
of
Hecke
type
if
it
admits
a
global
n
th
canonical
Hecke
correspondence
for
every
n
≥
1.
2
Proposition
3.5.
Suppose
that
there
exists
an
isomorphism
Ψ
log
:
X
log
→
(X
log
)
F
that
induces
the
morphism
Spf(Ψ
x
)
when
localized
at
every
supersingular
x
∈
X.
Then
X
log
is
of
Hecke
type.
Proof.
This
follows
immediately
from
gluing
together
Y
ord
with
the
various
Y
y
’s
by
means
of
Ψ
log
.
Corollary
3.6.
Suppose
that
X
log
,
along
with
all
of
its
supersingular
points
are
defined
over
F
p
2
.
Then
X
log
is
of
Hecke
type.
Proof.
Over
F
p
2
,
one
can
take
Ψ
log
to
be
the
identity,
for
(by
functoriality)
the
Ψ
x
’s
must
all
be
the
identity.
The
classical
example
of
a
case
where
X
log
is
of
Hecke
type
is
the
case
where
X
log
=
log
M
1,0
,
the
compactified
moduli
stack
of
elliptic
curves
(over
Z
p
).
This
case
is
studied
in
detail
in
[Shi],
Chapter
3.
To
see
that
the
supersingular
points
are
defined
over
F
p
2
,
one
2
repeats
the
argument
preceding
Proposition
3.2,
to
obtain
an
isomorphism
E
F
∼
=
E
for
every
supersingular
elliptic
curve
E.
Suppose
that
X
log
is
canonical
of
Hecke
type.
Then
we
remark
that
just
as
in
the
log
⊗N
classical
case,
one
can
define
Hecke
operators
on
H
0
(X,
(ω
X/S
)
)
Q
p
(as
well
as
one
the
log
étale
cohomology
of
X
Q
,
etc.).
Moreover,
(by
the
same
proofs
as
in
the
classical
case)
p
the
effect
on
q-expansions
(where
q
is
the
Serre-Tate
parameter
at
a
marked
point)
is
the
same
as
in
the
classical
case.
(See,
e.g.,
[Lang]
for
more
details.)
Next,
we
justify
the
assertion
(made
in
the
preceding
subsection)
that,
in
some
sense,
being
of
Hecke
type
is
a
rather
rare
phenomenon.
Indeed,
if
X
log
admits
a
global
first
canonical
Hecke
correspondence,
consider
its
reduction
modulo
p:
(D,
R)
F
p
:
Y
F
p
→
X
F
p
×
X
F
F
p
Let
Z
=
Y
F
p
.
Then
Z
=
Z
Z
,
with
Z
mapping
isomorphically
to
X
F
p
via
D
F
p
,
and
−1
D
F
p
|
Z
isomorphic
to
the
Frobenius
morphism
from
Z
∼
=
X
F
F
p
to
X
F
p
.
On
the
other
−1
hand,
R
F
p
must
map
Z
isomorphically
to
X
F
F
p
.
Thus,
Z
is
isomorphic
to
both
X
F
F
p
2
and
X
F
F
p
.
In
particular,
X
F
p
∼
=
X
F
F
p
,
i.e.,
the
moduli
of
X
F
p
are
defined
over
F
p
2
,
which
is
a
very
rare
phenomenon.
141
Remark.
At
the
present
time,
the
author
does
not
know
of
any
canonical
X
log
of
Hecke
type,
except
for
those
that
arise
directly
from
the
modular
case.
§4.
p-adic
Green’s
Functions
In
this
Section,
we
give
a
geometric
criterion
for
a
curve
to
be
canonical:
namely,
the
existence
of
a
Frobenius
lifting
of
the
right
height,
over
an
open
p-adic
formal
subscheme
of
the
curve,
with
“nice
behavior”
at
the
points
where
it
is
not
defined.
We
will
make
these
terms
precise
below,
but
the
point
of
interest
is
that
this
criterion
does
not
depend
ord
on
knowing
the
action
of
the
canonical
Frobenius
Φ
N
on
N
g,r
–
that
is,
it
is
intrinsic
to
the
curve
–
and,
moreover,
it
is
not
phrased
in
terms
of
indigenous
bundles.
Now
in
the
case
of
elliptic
curves
(regarded
parabolically),
the
canonical
lifting
defined
in
terms
of
indigenous
bundles
is
the
same
as
the
canonical
lifting
defined
in
Serre-Tate
theory
(one
of
the
definitions
of
which
is
the
existence
of
a
global
Frobenius
lifting).
Thus,
it
is
interesting
to
note
that
the
existence
of
an
“admissible
Frobenius
lifting”
amounts
to
just
the
existence
of
a
Frobenius
lifting
(in
the
case
of
elliptic
curves).
In
other
words,
one
may
regard
the
geometric
criterion
given
here
as
the
proper
hyperbolic
generalization
of
the
statement
that
an
elliptic
curve
(whose
reduction
modulo
p
is
ordinary)
is
Serre-Tate
canonical
if
and
only
if
it
admits
a
Frobenius
lifting.
Compactified
Frobenius
Liftings
In
this
subsection,
motivated
by
the
construction
of
the
pseudo-Hecke
correspondences
in
the
previous
Section,
we
define
the
general
notion
of
a
“compactified
Frobenius
lifting.”
Let
X
log
→
S
log
be
a
smooth
r-pointed
curve
of
genus
g.
Let
φ
log
:
Y
log
→
X
log
be
a
finite,
flat
morphism
such
that
Y
is
regular
(necessarily
of
dimension
two),
and
the
log
structure
on
Y
log
is
defined
by
a
divisor
on
Y
which
is
étale
over
A
and
equal
to
the
set-theoretic
inverse
image
of
the
divisor
of
marked
points
of
X.
Let
U
⊆
X
be
an
open
formal
subscheme
that
contains
all
the
marked
points.
We
endow
U
with
the
log
structure
pulled
back
from
X
log
,
and
call
the
resulting
log
formal
scheme
U
log
.
We
shall
use
the
notation
X
x
,
Y
y
,
etc.
(as
in
the
previous
subsection)
to
denote
the
formal
schemes
which
are
the
formal
neighborhoods
of
the
closed
points
x,
y,
etc.
Suppose
that
Φ
log
:
U
log
→
(U
log
)
F
is
a
Frobenius
lifting.
Definition
4.1.
We
shall
say
that
(φ
log
:
Y
log
→
X
log
;
ι)
is
a
naive
compactification
of
the
Frobenius
lifting
Φ
log
if,
when
we
take
V
=
φ
−1
(U
),
the
following
conditions
are
satisfied:
142
(1)
V
log
splits
as
a
disjoint
union
(V
log
)
(V
log
)
;
(2)
φ
log
|
V
:
(V
log
)
→
U
log
is
an
isomorphism;
(3)
ι
log
:
(V
log
)
→
(U
log
)
F
−1
(4)
φ
log
|
V
◦
(ι
log
)
−1
:
(U
log
)
F
is
an
isomorphism;
−1
→
U
log
is
the
morphism
(Φ
log
)
F
We
shall
frequently
identify
V
and
U
,
and
V
and
U
F
−1
−1
.
.
Suppose
that
(φ
log
;
ι
log
)
is
a
naive
compactification
for
Φ
log
.
Note
that
φ
is
necessarily
of
degree
p
+
1.
Also,
note
that
V
admits
a
canonical
embedding
ι
(1,0)
:
V
→
U
×
U
F
where
we
take
ι
(1,0)
=
(id,
Φ),
while
V
admits
an
embedding
ι
(0,1)
:
V
→
U
×
U
F
where
we
take
ι
(0,1)
=
((Φ)
F
−1
−1
,
id).
Let
x
∈
X(k).
Let
Ψ
x
:
R
x
F
2
∼
=
R
x
be
an
isomorphism.
Suppose
that
y
∈
Y
(k)
maps
to
x.
Let
(D
y
,
R
y
)
:
Y
y
→
X
x
×
X
x
F
be
a
closed
embedding,
where
D
y
is
obtained
by
restricting
φ.
def
Definition
4.2.
We
shall
say
that
H
x
=
(Ψ
x
;
(D
y
,
R
y
))
is
a
local
compactification
for
the
Frobenius
lifting
Φ
log
at
x
(relative
to
(φ
log
;
ι
log
))
if
(1)
y
is
the
unique
closed
point
of
Y
lying
over
x;
(2)
the
divisor
Y
y
→
X
x
×
X
x
F
is
symmetric
in
the
sense
that
the
two
divisors
that
it
induces
in
X
x
F
×
X
x
(by
switching
and
by
Frobenius-
conjugating,
then
applying
Ψ
x
)
are
the
same;
(3)
the
restriction
of
(D
y
,
R
y
)
to
V
is
the
union
of
ι
(1,0)
,
and
(ι
(0,1)
)
F
composed
with
Spf(Ψ
x
)
−1
×
(id).
143
2
Note
that
by
the
first
condition,
x
cannot
lie
in
U
.
Now
let
us
consider
(Y
y
)
Z/p
2
Z
.
Since
Y
y
⊆
X
x
×
X
x
F
,
it
follows
that
(Y
y
)
Z/p
2
Z
is
Spf
of
a
local
ring
of
the
form:
def
R
=
(A/p
2
A)[[ξ,
η]]/ψ(ξ,
η)
where
ξ
is
a
local
parameter
for
(X
x
)
Z/p
2
Z
,
and
η
=
ξ
F
is
the
Frobenius-conjugate
local
parameter
for
(X
x
F
)
Z/p
2
Z
.
Since
Y
y
→
X
x
is
flat,
the
last
condition
implies
that
(Y
y
)
F
p
has
exactly
two
irreducible
components,
both
of
which
are
reduced.
Thus,
if
we
denote
by
a
“bar”
the
reduction
of
functions
modulo
p,
we
see
that
ψ
is
a
product
of
two
distinct
prime
elements
of
k[[ξ,
η]].
In
fact,
we
can
say
more.
Outside
the
special
point
of
(Y
y
)
F
p
,
p
these
two
primes
define
the
closed
subschemes
ξ
−
η
and
η
p
−
Ψ
x
(ξ
f
=
ξ
p
−
η;
F
2
).
Let
2
g
=
η
p
−
Ψ
x
(ξ
F
)
Thus,
we
may
assume
that
ψ
=
f
·
g.
In
other
words,
we
can
write
ψ
=
f
·
g
+
π
where
π
∈
p
·
k[[ξ,
η]].
In
fact,
π
is
actually
p
times
a
unit
in
k[[ξ,
η]],
since
Y
y
is
regular.
So
far
we
have
been
working
with
functions
on
(Y
y
)
Z/p
2
Z
.
Now
let
us
restrict
to
functions
on
the
open
formal
subscheme
D(g)
⊆
(Y
y
)
Z/p
2
Z
(i.e.,
where
g
is
invertible).
Thus,
we
are
in
effect
restricting
to
the
graph
of
Φ.
Let
us
denote
the
restriction
morphism
on
functions
by
ζ
:
R
→
R[1/g].
Then
we
obtain,
in
R[1/g],
ζ(η)
=
ζ(ξ)
p
+
ζ(π)
·
ζ(g)
−1
By
interpreting
this
open
formal
subscheme
D(g)
as
the
graph
of
Φ,
this
tells
us
that
Φ
−1
(ξ
F
)
is
a
function
which
is
not
regular
at
x,
but
has
a
pole
of
order
one
(since
g
has
a
zero
of
order
one).
In
particular,
it
tells
us
that
the
Frobenius
lifting
Φ
does
not
admit
a
regular
extension
to
any
neighborhood
of
x.
We
summarize
this
as
follows:
Proposition
4.3.
If
H
x
is
a
local
compactification
of
Φ
log
at
x,
then
(Y
y
)
F
p
is
a
node,
and
x
∈
/
U
.
Also,
the
Frobenius
lifting
Φ
does
not
admit
an
extension
to
any
neighborhood
of
x.
Definition
4.4.
We
shall
call
C
=
(φ
log
;
ι
log
;
{H
x
}
x
∈U
/
)
144
a
compactification
of
the
Frobenius
lifting
Φ
log
if
(φ;
ι)
is
a
naive
compactification
of
Φ
log
,
and
for
each
x
∈
/
U
,
we
are
given
a
local
compactification
H
x
of
Φ
log
relative
to
(φ;
ι)
(where
k
in
the
definition
above
is
replaced
by
the
field
of
rationality
of
x).
Thus,
in
particular,
by
what
we
did
in
the
last
two
subsections,
Proposition
4.5.
Suppose
that
X
log
is
a
canonical
curve.
Then
its
first
canonical
pseudo-Hecke
correspondence
is
a
compactification
of
the
canonical
Frobenius
lifting
on
(X
log
)
ord
.
def
log
Suppose
that
(φ
log
;
ι
log
;
{H
x
}
x
∈U
.
Let
us
consider
Z
=
/
)
is
a
compactification
of
Φ
Y
F
p
.
It
follows
from
the
above
definition
that
Z
is
reduced
and
has
exactly
two
irreducible
components
Z
and
Z
with
V
F
p
⊆
Z
;
V
F
p
⊆
Z
.
Since
Z
is
geometrically
connected,
smooth,
proper,
and
birationally
equivalent
to
X
F
p
over
k,
it
follows
that
Z
∼
=
X
F
p
.
∼
F
−1
Similarly,
Z
=
X
F
p
.
Moreover,
except
at
the
points
of
intersection
of
Z
and
Z
(which
are
nodes),
Z
is
smooth
over
k.
Proposition
4.6.
(Assuming
that
X
log
is
hyperbolic)
Y
must
be
connected.
Proof.
It
suffices
to
prove
that
Z
is
connected.
Suppose
that
Z
is
not
connected.
Then
φ
Z
:
Z
→
X
F
p
is
finite
and
birational,
hence
an
isomorphism.
It
thus
follows
that
Z
lifts
to
a
connected
component
Y
of
Y
such
that
φ
Y
:
Y
→
X
is
an
isomorphism.
−1
On
the
other
hand,
Z
is
proper
and
smooth
over
k,
and
birational
to
X
F
F
p
,
hence
−1
Z
∼
=
X
F
,
and
φ|
Z
:
Z
→
X
F
is
the
Frobenius
morphism.
Moreover,
Z
lifts
to
a
F
p
p
connected
component
Y
of
Y
.
Thus,
φ|
Y
:
Y
→
X
is
a
Frobenius
lifting.
But
if
X
log
is
hyperbolic,
such
Frobenius
liftings
cannot
exist,
for
the
nonzero
morphism
of
line
bundles
log
→
ω
Y
log
/S
violates
degree
restrictions.
(φ|
Y
)
∗
ω
X/S
The
Height
of
a
Frobenius
Lifting
Finally,
we
note
that
often
it
is
useful
to
have
a
precise
measure
of
how
far
a
Frobenius
lifting
fails
to
extend
over
all
of
X.
For
this,
we
introduce
the
notion
of
the
height
of
a
Frobenius
lifting,
as
follows.
Let
F
→
X
F
p
be
the
T
F
p
-torsor
of
Frobenius
liftings
on
open
sub-log
schemes
of
Thus,
if
Φ
log
:
U
log
→
(U
log
)
F
is
a
Frobenius
lifting,
its
reduction
modulo
p
2
defines
a
section
σ
Φ
:
U
F
p
→
F
of
this
torsor.
Let
P
be
the
projective
bundle
that
canonically
compactifies
F.
Thus,
P
→
X
F
p
is
a
P
1
-bundle.
Recall
the
notion
of
the
canonical
height
of
a
section
of
P
→
X
F
p
,
introduced
at
the
beginning
of
Chapter
I,
§2.
Since
X
F
p
is
proper
over
k,
it
follows
that
σ
Φ
extends
uniquely
to
a
section
σ
Φ
:
X
F
p
→
P
.
We
now
make
the
following
log
X
Z/p
2
Z
.
145
Definition
4.7.
We
define
the
height
ht(Φ)
of
the
Frobenius
lifting
Φ
log
to
be
the
canonical
height
of
the
section
σ
Φ
of
P
→
X
F
p
.
More
concretely,
the
height
of
Φ
log
can
be
defined
as
follows.
If
x
∈
X
F
log
,
then
let
t
be
a
p
log
local
parameter
of
X
at
x.
Let
Ξ
x
be
a
local
Frobenius
lifting
defined
in
a
neighborhood
−1
of
x.
Then
p
1
(Ξ
−1
(t))
is
a
rational
function
δ
x
on
X
F
log
.
Let
us
say
that
the
local
x
(t)
−
Φ
p
log
height
ht
x
(Φ)
of
Φ
at
x
is:
(1)
equal
to
0
if
this
function
δ
x
is
regular
at
x;
(2)
equal
to
the
order
of
the
pole
of
δ
x
at
x
otherwise.
Then
we
have
the
formula:
Proposition
4.8.
We
have
p
[k(x)
:
k]
ht
x
(Φ)
ht(Φ)
+
(2g
−
2
+
r)
=
2
log
x∈X
F
p
Proof.
This
follows
immediately
from
considering
the
intersection
number
of
σ
Φ
with
the
“section
at
infinity”
given
by
the
complement
of
F
in
P
.
Corollary
4.9.
If
Φ
log
admits
a
compactification,
then
the
local
heights
at
points
outside
U
are
all
one.
Thus,
p
ht(Φ)
=
−
(2g
−
2
+
r)
+
deg
k
(X
−
U
)
F
p
2
where
we
regard
(X
−
U
)
F
p
as
having
the
reduced,
induced
scheme
structure.
Proof.
The
statement
about
local
heights
follows
from
the
explicit
computation
preceding
Proposition
4.3.
The
following
is
the
main
result
of
this
subsection:
Proposition
4.10.
If
Φ
log
has
height
≤
1−g−
12
r,
then
P
(with
its
connection
∇
P
induced
by
that
of
F)
is
a
nilpotent,
admissible
indigenous
bundle.
In
particular,
if
(P,
∇
P
)
is
also
ordinary,
then
X
log
is
isomorphic
to
a
canonical
curve
modulo
p
2
,
and
Φ
log
is
equal
to
the
canonical
Frobenius
lifting
(of
Theorem
3.1)
modulo
p
2
.
146
Proof.
Indeed,
suppose
that
ht(Φ)
≤
1
−
g
−
12
r.
Consider
the
Kodaira-Spencer
morphism
of
the
section
σ
Φ
of
P
→
X.
By
the
general
properties
of
FL-bundles
(Chapter
II,
§1),
we
know
that
the
Kodaira-Spencer
morphism
cannot
vanish
(for
the
section
at
infinity
of
P
→
X
is
the
unique
horizontal
section).
But
by
degree
considerations
(i.e.,
the
assumption
on
ht(Φ)),
once
the
Kodaira-Spencer
morphism
is
nonzero,
it
must
be
an
isomorphism.
It
thus
follows
that
P
(with
its
connection
induced
by
that
of
F)
is
a
nilpotent,
admissible
indigenous
bundle.
The
last
statement
follows
from
the
construction
of
the
canonical
lifting
and
the
canonical
Frobenius.
Thus,
we
see
that
the
compactified
Frobenius
liftings
that
we
are
really
interested
in
are
the
ones
that
“look
nice
modulo
p:”
Definition
4.11.
A
compactified
Frobenius
lifting
C
=
(φ
log
;
ι
log
;
{H
x
}
x
∈U
/
)
is
called
admissible
if
(1)
ht(Φ)
=
g
−
1
+
12
r;
(2)
the
associated
(P,
∇
P
)
(as
in
Proposition
4.10)
is
ordinary;
(3)
the
reductions
modulo
p
of
the
isomorphisms
Ψ
x
(that
make
up
H
x
)
are
equal
to
the
canonical
“Ψ
x
”
of
Definition
3.3.
Note
that
for
an
admissible
compactified
Frobenius
C,
all
the
objects
involved
(that
is,
Φ
log
;
φ
log
:
Y
log
→
X
log
;
ι
log
;
Ψ
x
;
(D
y
,
R
y
))
are
completely
determined
modulo
p
(up
to
isomorphism)
once
one
fixes
the
supersingular
divisor
(X
−
U
)
F
p
.
Or,
in
other
words,
Proposition
4.12.
An
admissible
compactified
Frobenius
C
on
X
log
determines
a
p-adic
quasiconformal
equivalence
class
α
to
which
X
log
belongs.
If
two
admissible
compactified
Frobenii
C
and
C
on
X
log
determine
the
same
α,
then,
modulo
p,
all
the
objects
that
make
up
C
are
isomorphic
to
those
that
make
up
C
.
Admissible
Frobenius
Liftings
Since
an
admissible
compactified
Frobenius
is
determined
modulo
p
by
the
p-adic
quasiconformal
equivalence
class
α,
the
next
step
is
to
understand
what
the
possible
de-
formations
looks
like.
Let
C
be
an
admissible
compactified
Frobenius,
and
let
us
consider
C
Z/p
2
Z
,
i.e.,
the
reductions
modulo
p
2
of
all
the
objects
involved.
Suppose
we
start
with
147
the
data
(Φ
log
)
Z/p
2
Z
;
(Ψ
x
)
Z/p
2
Z
(for
all
supersingular
x).
Now
it
is
easy
to
see
(by
looking
at
points
valued
in
an
arbitrary
scheme)
that
the
symmetry
condition
on
the
divisor
Y
y
amounts
to
the
statement
that
(Ψ
x
)
Z/p
2
Z
restricted
to
the
ordinary
locus
commutes
with
Φ
Z/p
2
Z
.
Let
β
be
an
automorphism
of
(X
x
)
Z/p
2
Z
which
is
equal
to
the
identity
modulo
p.
Then
we
claim
that
if
Φ
Z/p
2
Z
commutes
with
β,
then
β
is
the
identity.
Indeed,
since
derivations
on
X
F
p
act
trivially
on
functions
that
are
p
th
powers,
we
get
that
Φ
Z/p
2
Z
◦
β
=
Φ
Z/p
2
Z
Thus,
β
F
◦
Φ
Z/p
2
Z
=
Φ
Z/p
2
Z
which
implies
that
β
is
the
identity,
since
Φ
Z/p
2
Z
is
faithfully
flat.
Thus,
in
summary,
the
(Ψ
x
)
Z/p
2
Z
are
determined
uniquely
by
the
condition
that
they
commute
with
Φ
Z/p
2
Z
.
Next,
we
consider
Y
log
.
We
can
break
Y
log
up
into
three
parts:
(V
log
)
;
(V
log
)
;
and
the
Y
y
’s.
Since
(V
log
)
and
(V
log
)
are
determined
up
to
natural
isomorphism
by
X
log
,
it
remains
to
determine
the
Y
y
’s,
and
the
gluing
morphisms.
But
Y
y
and
its
gluing
morphisms
to
(V
log
)
and
(V
log
)
are
completely
specified
once
one
knows
the
divisor
Y
y
in
X
x
×X
X
F
.
Moreover,
it
follows
from
the
condition
(3)
of
Definition
4.2,
that
this
divisor
is
determined
by
Φ
and
Ψ
x
.
But
we
just
saw
that
(Ψ
x
)
Z/p
2
Z
is
determined
by
Φ
Z/p
2
Z
.
Thus,
we
conclude
2
that
(H
x
)
Z/p
2
Z
and
φ
log
Z/p
2
Z
are
completely
determined
by
Φ
Z/p
Z
.
Sorting
through
all
the
definitions,
we
thus
see
that
we
have
proven
that
C
Z/p
2
Z
is
entirely
determined
by
the
p-adic
quasiconformal
equivalence
class
α
and
the
deformation
Φ
Z/p
2
Z
.
Moreover,
there
is
nothing
special
about
working
modulo
p
2
:
the
same
arguments
can
be
made
modulo
an
arbitrary
power
of
p.
Thus,
we
see
that
we
have
proven
the
following
result:
Lemma
4.13.
Let
C
be
an
admissible
compactified
Frobenius
lifting
on
X
log
.
Then
C
Z/p
n
Z
is
completely
determined
by
Φ
log
Z/p
n
Z
.
This
Lemma
suggests
the
following
definition:
Definition
4.14.
Let
α
be
a
p-adic
quasiconformal
equivalence
class
to
which
X
log
belongs.
Let
U
⊆
X
be
the
ordinary
locus
for
α.
Let
Φ
log
:
U
log
→
(U
log
)
F
be
a
Frobenius
lifting
over
U
log
.
Then
we
shall
say
that
Φ
log
is
an
admissible
Frobenius
lifting
for
(X
log
,
α)
if
it
arises
from
a
(necessarily
unique)
admissible
compactified
Frobenius
C.
Thus,
we
can
can
regard
admissible
Frobenius
liftings
as
being
Frobenius
liftings
(over
the
ordinary
locus)
that
happen
to
have
special
behavior
near
the
supersingular
points.
148
Next
let
us
consider
two
admissible
Frobenius
liftings
Φ
log
and
(Φ
log
)
on
the
same
curve
X
log
.
Let
us
suppose
that
they
are
equal
modulo
p
n
(where
n
≥
2).
In
this
discussion,
we
shall
always
be
working
modulo
p
n+1
,
so
(by
abuse
of
notation)
we
shall
use
Φ
log
and
(Φ
log
)
to
denote
the
respective
reductions
modulo
p
n+1
.
Now
just
as
in
the
discussion
preceding
Proposition
4.3,
for
Φ
log
,
(Y
y
)
Z/p
n+1
Z
is
Spf
of
a
local
ring
of
the
form:
def
R
=
(A/p
n+1
A)[[ξ,
η]]/ψ(ξ,
η)
where
ξ
is
a
local
parameter
for
(X
x
)
Z/p
n+1
Z
,
and
η
=
ξ
F
is
the
Frobenius-conjugate
local
parameter
for
(X
x
F
)
Z/p
n+1
Z
.
Moreover,
we
may
assume
that
ψ
=
f
·
g
+
π
where
f
=
ξ
p
−
η;
2
g
=
η
p
−
Ψ
x
(ξ
F
)
and
π
∈
p
·
(A/p
n
A)[[ξ,
η]].
In
fact,
π
is
actually
p
times
a
unit,
since
Y
y
is
regular.
Similarly,
for
(Φ
log
)
,
we
have
R
=
(A/p
n+1
A)[[ξ,
η]]/ψ
(ξ,
η)
def
with
ψ
=
f
·
g
+
π
,
and
g
=
η
p
−
Ψ
x
(ξ
F
).
Thus,
2
g
≡
g
(mod
p
n
);
π
≡
π
(mod
p
n
)
So
far
we
have
been
working
with
functions
on
(Y
y
)
Z/p
n+1
Z
.
Now
let
us
restrict
to
the
open
subscheme
corresponding
to
the
graph
of
Φ
(or
Φ
).
Let
us
denote
the
restriction
morphism
on
functions
by
ζ.
Then
we
obtain,
on
this
open
subscheme,
in
the
unprimed
case:
ζ(η)
=
ζ(ξ)
p
+
ζ(π)
·
ζ(g)
−1
and,
in
the
primed
case:
ζ(η)
=
ζ(ξ)
p
+
ζ(π
)
·
ζ(g
)
−1
=
ζ(ξ)
p
+
ζ(π
)
·
ζ(g)
−1
since
the
difference
between
g
and
g
becomes
zero
when
multiplied
by
p.
Ultimately,
we
are
interested
in
computing
the
difference
between
the
two
Frobenius
liftings
Φ
and
Φ
.
149
That
is,
we
wish
to
understand
the
difference
between
where
η
is
taken
by
the
two
liftings.
But
by
the
above
formulas,
the
difference
is
of
the
form:
{ζ(π)
−
ζ(π
)}(ζ(g))
−1
Moreover,
because
the
difference
in
brackets
is
divisible
by
p
n
,
only
the
residue
of
g
modulo
p
is
involved
in
the
above
expression.
Since
this
residue
has
a
zero
of
order
exactly
one
at
x,
it
follows
that
the
difference
between
the
two
Frobenius
liftings
–
which
forms
a
section
of
T
F
p
over
X
ord
–
has
poles
of
order
at
most
one
at
the
supersingular
points.
Since
the
morphism
H
T
F
p
:
T
F
p
→
(τ
X
log
/S
log
)
F
p
(given
by
composing
the
p-curvature
of
(P,
∇
P
)
with
the
projection
given
by
the
Hodge
filtration)
has
zeroes
at
the
supersingular
points,
it
thus
follows
that
the
difference
between
Φ
log
and
(Φ
log
)
defines
a
global
section
of
(τ
X
log
/S
log
)
F
p
.
Since
we
are
dealing
with
hy-
perbolic
curves,
though,
(τ
X
log
/S
log
)
F
p
has
negative
degree,
hence
has
no
global
sections.
Thus,
Φ
log
=
(Φ
log
)
.
In
summary,
we
have
proven
the
following
strengthened
form
of
Lemma
4.13:
Lemma
4.15.
If
there
exists
an
admissible
Frobenius
lifting
of
(X
log
,
α),
then
it
is
unique,
and
contains
no
nontrivial
deformations
modulo
any
power
of
p.
Geometric
Criterion
for
Canonicality
Let
us
fix
X
log
→
S
log
,
a
smooth
r-pointed
curve
of
genus
g,
and
a
p-adic
quasicon-
formal
equivalence
class
α
to
which
X
log
belongs.
Let
Φ
log
be
an
admissible
Frobenius
lifting
for
(X
log
,
α).
log
→
S
log
is
also
a
smooth
r-pointed
curve
of
genus
g
such
that
Now
suppose
that
X
log
(
X
)
F
p
=
X
F
p
.
The
following
“Rigidity
Lemma”
is
fundamental
to
this
subsection:
log
log
,
α)
admits
an
admissible
Frobenius
lifting
(Φ
log
)
.
Then
Lemma
4.16.
Suppose
that
(
X
log
=
X
log
.
X
Proof.
We
propose
to
prove
inductively
(on
n)
the
following
statement:
log
,
(Φ
log
)
)
coincides
with
(X
log
,
Φ
log
)
modulo
p
n+1
.
(*)
(
X
We
know
that
this
statement
holds
for
n
=
0.
Assume
that
it
holds
for
n
−
1.
Consider
log
log
log
the
difference
between
the
deformations
X
Z/p
n+1
Z
and
X
Z/p
n+1
Z
of
X
Z/p
n
Z
.
It
defines
a
class
150
μ
∈
H
1
(X
F
p
,
(τ
X
log
/S
log
)
F
p
)
Let
C
(respectively,
C
)
be
the
compactified
Frobenius
corresponding
to
Φ
log
(respectively,
log
log
)
Z/p
n
Z
,
we
can
consider
the
difference
between
the
defor-
(Φ
log
)
).
Since
Y
Z/p
n
Z
=
(Y
log
log
log
mations
Y
Z/p
)
Z/p
n+1
Z
of
Y
Z/p
n
Z
.
This
gives
us
a
class
n+1
Z
and
(Y
ν
∈
Ext
1
Y
F
p
(Ω
Y
log
/k
,
O
Y
F
p
)
F
p
Since
φ
Z/p
n
Z
=
φ
Z/p
n
Z
,
the
pull-back
map
defined
by
either
of
these
morphisms
gives
us
log
a
map
φ
−1
:
φ
∗
(ω
X/S
)
F
p
→
Ω
Y
log
/k
,
which
is
generically
zero
over
the
component
of
Y
F
p
F
p
that
we
called
Z
in
the
proof
of
Proposition
4.6.
Now
φ
−1
induces
a
pull-back
morphism
on
global
Ext’s
ψ
:
Ext
1
Y
F
p
(Ω
Y
log
/k
,
O
Y
F
p
)
→
H
1
(Y
F
p
,
φ
∗
(τ
X
log
/S
log
)
F
p
)
F
p
The
condition
that
the
morphism
φ
Z/p
n+1
Z
deform
compatibly
with
the
deformation
of
log
,
(Y
)
log
)
Z/p
n+1
Z
to
a
morphism
φ
n+1
is
exactly
that
(X
log
,
Y
log
)
Z/p
n+1
Z
to
(
X
Z
Z/p
ψ(ν)
=
μ|
Y
F
p
Note
that
this
condition,
as
well
as
the
cohomology
modules
in
which
μ
and
ν
live,
log
log
are
independent
of
n.
Thus,
by
adding
μ
and
ν
to
φ
log
Z/p
2
Z
:
Y
Z/p
2
Z
→
X
Z/p
2
Z
,
we
obtain
a
new
morphism
φ
log
:
Y
log
→
X
log
(of
Z/p
2
Z-flat
schemes).
Then,
by
restricting
φ
log
to
the
open
subscheme
of
Y
log
defined
by
the
ordinary
locus
of
Z
,
we
obtain
a
Frobenius
lifting
Ξ
log
on
the
ordinary
locus
of
X
log
.
The
only
points
at
which
Ξ
is
not
defined
are
the
supersingular
points
(determined
by
α).
Moreover,
by
the
calculation
of
the
discussion
preceding
Proposition
4.3,
it
follows
that
the
local
height
of
Ξ
log
at
a
supersingular
point
is
≤
1.
Indeed,
in
the
notation
of
loc.
cit.,
“π”
is
equal
to
p
times
an
element
of
k[[ξ,
η]],
which,
this
time,
might
not
be
a
unit
since
Y
log
might
not
be
the
reduction
modulo
p
2
of
a
regular
scheme;
hence
the
inequality
≤
1,
rather
than
the
sharp
equality
=
1.
At
any
rate,
it
thus
follows
that
ht(Ξ)
≤
ht(Φ
Z/p
2
Z
).
But
then,
by
Proposition
4.10,
X
log
is
equal
to
some
canonical
curve
reduced
modulo
p
2
,
and,
by
Proposition
2.6,
(4),
of
Chapter
II,
it
log
thus
follows
that
this
canonical
curve
is
the
one
determined
by
α.
Thus,
X
log
=
X
Z/p
2
Z
.
log
log
n+1
and
X
coincide
modulo
p
.
By
Lemma
4.15
But
this
means
that
μ
=
0,
so
X
log
(the
rigidity
of
an
admissible
Frobenius
lifting),
it
thus
follows
that
Φ
and
(Φ
log
)
also
coincide
modulo
p
n+1
.
This
proves
the
induction
step,
and
hence
the
Lemma.
Putting
everything
together,
we
see
that
we
have
proven
the
following
geometric
criterion
for
a
curve
to
be
canonical:
151
Theorem
4.17.
Let
X
log
→
S
log
be
a
smooth
r-pointed
curve
of
genus
g.
Let
α
be
a
p-adic
quasiconformal
equivalence
to
which
X
log
belongs.
Then
X
log
is
canonical
if
and
only
if
(X
log
,
α)
admits
an
admissible
Frobenius
lifting.
Proof.
We
saw
in
§3
that
a
canonical
curve
admits
an
admissible
Frobenius
lifting.
On
the
other
hand,
given
an
(X
log
,
α)
which
admits
an
admissible
Frobenius
lifting,
there
exists
log
≡
X
log
modulo
p)
which
is
canonical,
hence
admits
an
admissible
log
,
α)
(with
X
an
(
X
log
.
Frobenius
lifting.
Thus,
by
Lemma
4.16,
it
follows
that
X
log
=
X
Definition
4.18.
Suppose
that
(X
log
,
α)
admits
an
admissible
Frobenius
lifting
Φ
log
.
Then
we
shall
call
Φ
log
the
p-adic
Green’s
function.
Remark.
The
justification
for
this
terminology
is
as
follows.
In
the
classical
complex
case,
one
of
the
main
approaches
to
proving
that
hyperbolic
curves
can
be
uniformized
by
the
upper
half
plane
is
given
by
constructing
a
Green’s
function
on
the
universal
covering
space
of
the
Riemann
surface
(see,
e.g.,
[FK]).
Once
one
proves
that
the
universal
covering
space
is
just
the
upper
half
plane,
then
one
sees
that
this
Green’s
function
is
really
just
the
logarithm
of
the
hyperbolic
distance
between
two
points.
On
the
other
hand,
the
canonical
Frobenius
lifting
Φ
log
may
also
be
regarded
as
giving
us
a
notion
of
distance
on
X
log
.
Indeed,
in
the
classical
modular
case,
where
X
log
parametrizes
elliptic
curves,
if
one
can
get
from
point
a
to
point
b
by
applying
Φ
log
a
total
of
N
times,
then
it
means
that
the
corresponding
elliptic
curves
are
isogenous
via
a
cyclic
isogeny
of
order
p
N
.
Thus,
the
analogy
between
Φ
log
and
the
classical
complex
Green’s
function
(which
is
just
the
logartihm
of
the
hyperbolic
distance)
will
be
established
once
one
accepts
that
isogeny
is
the
proper
analogue
of
distance.
But
to
see
this,
one
need
merely
think
of
lattices
in
Q
2
p
,
which
one
can
draw
schematically
as
a
graph.
Then
two
lattices
are
related
by
an
isogeny
of
order
p
N
if
and
only
if
they
are
N
edges
apart
on
this
graph.
This
establishes
the
relationship
between
isogeny
and
distance.
Remark.
We
also
observe
that
for
elliptic
curves
(regarded
parabolically),
the
same
defi-
nition
of
compactified
Frobenius
liftings,
and
admissible
Frobenius
liftings
goes
through,
but
everything
is
trivial,
since
there
are
no
supersingular
points
to
contend
with.
Thus,
we
(trivially)
obtain
the
analogue
of
Theorem
4.17:
that
an
elliptic
curve
over
A
with
ordinary
reduction
is
canonical
if
and
only
if
it
admits
a
Frobenius
lifting
defined
everywhere
on
the
curve.
152
Chapter
V:
Uniformizations
of
Ordinary
Curves
§0.
Introduction
Having
studied
the
case
of
canonical
curves
in
the
previous
Chapter,
in
this
Chapter
we
turn
to
the
case
of
arbitrary
curves
with
ordinary
reduction
modulo
p.
We
do
this
by
working
with
the
universal
case:
i.e.,
the
universal
curve
over
the
moduli
stack.
Unlike
the
canonical
case,
one
does
not
quite
obtain
such
objects
as
the
canonical
Galois
rep-
resentation
or
the
canonical
log
p-divisible
group
over
the
given
base.
Instead,
one
must
pass
to
various
“schemes
of
multiplicative
periods”
–
i.e.,
certain
infinite
coverings
of
the
original
base
–
in
order
to
obtain
such
objects.
On
the
other
hand,
since
these
objects
are
canonically
associated
to
the
curve
over
the
given
base,
it
is
natural
to
guess
that
they
should
descend
from
the
scheme
of
multiplicative
periods
back
down
to
the
original
base
in
some
appropriate
sense.
The
key
idea
here
is
that,
for
instance
in
the
case
of
the
canonical
Galois
representation
(which
is
fundamental
to
the
construction
of
all
the
other
objects),
if
one
works
with
modules
of
rank
two,
not
over
Z
p
,
but
over
some
appropriate
ring
of
p-adic
periods
D
Gal
,
then
one
can
in
fact
construct
a
canonical
Galois
representation
over
the
original
base.
Thus,
one
obtains
a
representation
of
the
entire
arithmetic
fundamental
Gal
group
into
GL
±
),
which
in
some
sense
extends
the
representation
of
the
geometric
2
(D
fundamental
group
into
GL
±
2
(Z
p
).
Moreover,
(in
the
hyperbolic
case)
this
representation
Gal
)
is
canonical,
and
dual
crystalline
in
of
the
arithmetic
fundamental
group
into
GL
±
2
(D
some
appropriate
sense,
despite
the
fact
that
(unlike
the
case
handled
in
[Falt],
§2),
it
is
on
a
space
of
infinite
rank
over
Z
p
.
The
process
of
passing
from
the
canonical
representa-
tion
of
the
geometric
fundamental
group
into
GL
±
2
(Z
p
)
to
the
canonical
representation
of
Gal
the
arithmetic
fundamental
group
into
GL
±
(D
)
is
a
sort
of
crystalline
analogue
of
the
2
notion
of
an
induced
representation
in
group
theory.
We
therefore
refer
to
this
process
as
the
process
of
crystalline
induction.
Once
one
has
this
canonical
dual
crystalline
representation
of
the
arithmetic
funda-
Gal
mental
group
into
GL
±
),
one
can
linearize
the
obstruction
to
extending
the
repre-
2
(D
sentation
of
the
geometric
fundamental
group
into
GL
±
2
(Z
p
)
to
the
full
arithmetic
funda-
mental
group.
This
linearization
tells
one,
for
instance,
that
as
soon
as
one
can
extend
the
representation
of
the
geometric
fundamental
group
into
GL
±
2
(Z
p
)
at
all
to
the
arith-
metic
fundamental
group,
this
extension
is
automatically
dual
crystalline.
This
procedure
of
linearizing
the
obstruction
also
allows
one
to
see
that
this
obstruction
is
precisely
the
hyperbolic
analogue
of
the
obstruction
to
splitting
a
certain
exact
sequence
of
p-adic
local
systems
on
the
moduli
stack
of
ordinary
elliptic
curves
(in
the
parabolic
case).
§1.
Crystalline
Induction
In
Chapter
III,
we
constructed
a
Frobenius-invariant
indigenous
bundle
on
the
univer-
ord
sal
curve
over
N
g,r
.
Unfortunately,
unlike
the
case
of
a
canonical
curve,
such
information
153
does
not
immediately
constitute
an
object
of
the
category
MF
∇
(see
[Falt],
§2),
so
we
cannot
immediately
convert
it
into
a
Galois
representation.
The
problem
is
that
our
con-
nection
on
the
indigenous
bundle
is
only
a
relative
connection
(for
the
universal
curve
over
ord
N
g,r
),
not
a
full
connection
on
the
total
space
of
the
universal
curve.
Also,
the
obstruction
to
extending
it
to
a
full
connection
on
the
total
space
of
the
universal
curve
is
nonzero.
Thus,
in
order
to
obtain
a
Galois
representation,
we
must
replace
the
indigenous
bundle
by
a
certain
natural
“thickening”
of
the
indigenous
bundle.
This
thickening
formally
car-
ries
the
structure
of
an
object
of
the
category
MF
∇
,
but
has
the
disadvantage
of
being
of
infinite
rank,
so
that
we
cannot
immediately
apply
the
theory
of
[Falt]
to
this
object.
Fortunately,
it
is
not
difficult
to
extend
the
theory
of
[Falt]
so
as
to
handle
such
objects
of
infinite
rank.
We
thus
obtain
a
Galois
representation,
as
desired,
which
turns
out
to
be
a
sort
of
crystalline
analogue
of
the
notion
of
an
“induced
representation”
in
group
theory.
The
Crystalline-Induced
MF
∇
-object
def
Let
p
be
an
odd
prime.
Let
S
be
formally
smooth
over
A
=
W
(k),
where
k
is
a
perfect
field
of
characteristic
p.
Let
us
assume
that
S
is
endowed
with
a
log
structure
induced
by
a
relative
divisor
with
normal
crossings
over
W
(k).
Let
S
log
be
the
resulting
log
formal
scheme.
Let
f
log
:
X
log
→
S
log
be
an
r-pointed
stable
curve
of
genus
g.
Also,
let
us
assume
that
the
classifying
morphism
S
→
M
g,r
defined
by
f
log
is
étale.
Let
Φ
log
:
S
log
→
S
log
be
an
ordinary
Frobenius
lifting.
Now
let
us
suppose
that
our
indigenous
bundle
(E,
∇
E
)
on
X
log
is
invariant
under
the
renormalized
Frobenius
(Chapter
III,
Definition
1.4):
that
is,
Φ
E
:
(E,
∇
E
)
∼
=
F
∗
S
log
(E,
∇
E
)
Φ
where
the
superscript
“Φ”
denotes
pull-back
by
Φ,
and
the
subscript
“S
log
”
denotes
that
we
are
considering
the
relative
renormalized
Frobenius
pull-back
over
S
log
.
(We
shall
denote
by
F
∗
A
the
renormalized
Frobenius
pull-back
over
A.)
Let
D
S
be
the
quasi-coherent
O
S
-algebra
(with
the
O
S
-action
from
the
right)
which
is
obtained
by
taking
the
p-adic
completion
of
the
PD-envelope
of
the
diagonal
embedding
of
S
log
in
S
log
×
A
S
log
(see,
e.g.,
[Kato],
§5.8).
Thus,
D
S
has
an
ideal
I
S
⊆
D
S
with
D
S
/I
S
=
log
log
O
S
,
and
I
S
/I
S
2
∼
=
Ω
S/A
=
Ω
S
.
Note
that
D
S
has
a
natural
logarithmic
connection
∇
D
S
.
We
shall
regard
D
S
as
a
filtered
object
with
connection,
whose
filtration
is
given
by
[i]
F
i
(D
S
)
=
I
S
(i.e.,
divided
powers
of
I
S
).
Thus,
the
Kodaira-Spencer
morphism
for
the
log
subquotient
of
the
filtration
given
by
F
1
(D
S
)/F
2
(D
S
)
=
I
S
/I
S
2
∼
=
Ω
S
is
the
identity
map.
Note
that
the
Frobenius
lifting
Φ
log
on
S
log
induces
a
morphism
Φ
D
S
:
Φ
∗
D
S
→
D
S
which
preserves
the
Hodge
filtration.
Finally,
let
us
denote
by
((D
S
)
X
,
∇
(D
S
)
X
)
the
pull-back
of
(D
S
,
∇
D
S
)
to
X
log
.
Next,
let
us
consider
the
obstruction
to
defining
a
full
logarthmic
connection
∇
on
E
(i.e.,
relative
to
X
log
→
Spec(A))
with
the
following
properties:
154
(1)
∇
has
trivial
determinant;
(2)
the
restriction
of
∇
to
a
relative
connection
(for
X
log
→
S
log
)
is
∇
E
;
(i.e.,
the
(3)
the
curvature
of
∇
is
an
Ad(E)-valued
section
of
∧
2
Ω
log
S
log
log
ω
X/S
⊗
O
S
Ω
S
-part
of
the
curvature
vanishes).
It
is
easy
to
see
that
the
obstruction
class
to
defining
such
a
connection
is
a
section
η
E
of
log
0
1
R
1
f
DR,∗
Ad(E)
⊗
O
S
Ω
log
S
=
F
(R
f
DR,∗
Ad(E)
⊗
O
S
Ω
S
)
log
and
Φ
E
whose
projection
to
R
1
f
∗
τ
X
log
/S
log
⊗
O
S
Ω
log
S
is
the
identity.
Also,
note
that
Φ
log
1
induce
a
Frobenius
action
on
R
f
DR,∗
Ad(E)
⊗
O
S
Ω
S
,
which,
by
naturality,
preserves
η
E
.
Thus,
in
particular,
we
see
that
unless
we
modify
E
in
some
way,
there
is
no
hope
of
constructing
a
full
connection
∇
as
specified
above.
PD
Thus,
we
make
the
following
construction.
Let
us
write
S
log
×
S
log
for
Spf(D
S
)
(where
we
take
“Spf”
with
respect
to
the
p-adic
topology).
Similarly,
we
shall
write
PD
PD
PD
S
log
×
S
log
×
S
log
,
X
log
×
X
log
,
etc.
for
the
obvious
p-adic
completions
of
PD-
PD
envelopes
at
the
respective
diagonals.
Also,
we
have
two
projections
π
R
,
π
L
:
S
log
×
S
log
→
S
log
to
the
left
and
right
by
which
we
can
pull-back
X
log
→
S
log
to
obtain
curves
PD
PD
(X
log
)
L
→
S
log
×
S
log
,
and
(X
log
)
R
→
S
log
×
S
log
.
Moreover,
both
of
these
curves
PD
form
PD-thickenings
of
X
log
→
S
log
→
S
log
×
S
log
(where
the
second
morphism
is
the
diagonal
embedding).
It
thus
follows
that
if
we
pull-back
(E,
∇
E
)
to
obtain
an
indigenous
PD
bundle
on
the
curve
(X
log
)
L
→
S
log
×
S
log
,
this
indigenous
bundle
defines
a
crystal
PD
on
Crys(X
log
/(S
log
×
S
log
))
which
we
can
then
evaluate
on
the
thickening
(X
log
)
R
to
obtain
a
rank
two
vector
bundle
E
(on
X
R
).
If
we
then
push
this
sheaf
E
forward
via
the
projection
X
R
→
X,
we
obtain
a
quasi-coherent
sheaf
E
D
on
X.
Moreover,
E
D
has
the
structure
of
a
D
S
-module,
hence
of
a
(D
S
)
X
-module.
In
fact,
E
D
is
a
locally
free
(D
S
)
X
-
module
of
rank
two.
Moreover,
E
D
is
equipped
with
a
natural
Hodge
filtration
compatible
with
that
of
(D
S
)
X
.
Next,
we
would
like
to
equip
E
D
with
a
full
logarithmic
connection
that
is
compatible
with
its
structure
as
a
(D
S
)
X
-module
and
the
connection
on
(D
S
)
X
.
First,
note
that
PD
PD
PD
PD
X
log
×
S
log
×
S
log
→
S
log
×
S
log
×
S
log
is
a
PD-thickening
of
X
log
→
S
log
→
PD
PD
S
log
×
S
log
×
S
log
(where
the
second
morphism
is
the
diagonal
embedding).
Thus,
if
we
PD
PD
PD
pull-back
(E,
∇
E
)
to
X
log
×
S
log
×
S
log
,
we
obtain
a
crystal
E
on
Crys(X
log
/(S
log
×
PD
PD
PD
PD
PD
S
log
×
S
log
)).
On
the
other
hand,
S
log
×
X
log
×
X
log
→
S
log
×
S
log
×
S
log
is
also
PD
PD
a
PD-thickening
of
X
log
→
S
log
→
S
log
×
S
log
×
S
log
.
Thus,
if
we
evaluate
E
on
PD
PD
PD
PD
S
log
×
X
log
×
X
log
,
and
then
push
forward
via
the
projection
S
log
×
X
log
×
X
log
→
155
PD
PD
X
log
×
X
log
,
we
obtain
a
sheaf
E
on
X
log
×
X
log
.
Now
since
(D
S
)
X
is
equipped
PD
with
a
connection,
the
two
pull-backs
of
(D
S
)
X
to
X
log
×
X
log
via
the
two
projections
PD
X
π
R
,
π
L
X
:
X
log
×
X
log
→
X
log
can
be
identified;
we
denote
the
resulting
sheaf
of
algebras
PD
on
X
log
×
X
log
by
(D
S
)
PD
X
log
×
X
log
.
Then
E
is
equipped
with
the
structure
of
a
locally
free
-module
of
rank
two.
On
the
other
hand,
from
the
definition
of
E
D
,
it
follows
PD
X
log
×
X
log
X
∗
that
both
(π
L
X
)
∗
E
D
and
(π
R
)
E
D
are
naturally
isomorphic
(as
(D
S
)
-modules)
to
PD
X
log
×
X
log
PD
X
∗
E
,
hence
to
each
other.
This
isomorphism
(π
L
X
)
∗
E
D
∼
)
E
D
on
X
log
×
X
log
defines
=
(π
R
a
full
logarithmic
connection
∇
E
D
on
E
D
(with
respect
to
X
log
→
Spec(A)).
Moreover,
(D
S
)
one
checks
easily
that
this
connection
is
integral.
Finally,
we
have
a
Frobenius
action
Φ
E
D
:
F
∗
S
log
(E
D
⊗
D
S
,Φ
D
S
D
S
)
∼
=
E
D
Here,
in
the
definition
of
F
∗
S
log
,
we
first
pull
back
the
relevant
crystal
by
means
of
relative
Frobenius,
and
then
consider
the
subsheaf
consisting
of
sections
whose
reduction
modulo
p
is
contained
in
the
subsheaf
of
Φ
∗
X
F
p
(E
D
⊗
D
S
,Φ
D
S
D
S
)
F
p
∼
=
(Φ
∗
X
F
p
E
F
p
)
⊗
O
S
D
S
given
by
(Φ
∗
X
F
p
F
1
(E)
F
p
)
⊗
O
S
D
S
⊆
(Φ
∗
X
F
p
E
F
p
)
⊗
O
S
D
S
Theorem
1.1.
Over
X
log
,
there
exists
a
natural,
locally
free,
rank
two
(D
S
)
X
-module
E
D
equipped
with
a
Hodge
filtration,
a
full
integrable
logarithmic
connection
∇
E
D
(relative
to
X
log
→
Spec(A)),
and
a
Frobenius
action
Φ
E
D
:
F
∗
S
log
(E
D
⊗
D
S
,Φ
D
S
D
S
)
∼
=
E
D
such
that
E
D
⊗
D
S
(D
S
/I
S
)
=
E.
The
Ring
of
Additive
Periods
Before
we
can
convert
the
induced
object
of
Theorem
1.1
into
a
Galois
representation,
we
must
first
study
the
Galois
representation
associated
to
D
S
,
with
its
natural
filtration,
connection,
and
Frobenius
action.
Once
we
have
done
this,
since
E
D
is
of
finite
rank
over
156
D
S
,
converting
E
D
into
a
Galois
representation
will
be
no
more
difficult
than
the
“classical
case”
discussed
in
[Falt],
§2.
Let
us
first
note
that,
just
as
when
we
constructed
“canonical
affine
coordinates”
in
Chapter
III,
§1,
by
considering
the
slopes
of
the
Frobenius
action
Φ
D
S
,
we
obtain
a
unique
Frobenius-equivariant
embedding
of
O
S
-modules
Ω
log
S
→
I
S
whose
composite
with
the
projection
I
S
→
I
S
/I
S
2
=
Ω
log
S
is
the
identity.
It
is
here
that
we
use
the
divided
powers
of
I
S
⊆
D
S
.
Let
us
write
Geo(D
S
)
for
the
subbundle
of
D
S
generated
by
Ω
log
S
and
O
S
.
Note
that
Geo(D
S
)
is
stabilized
by
∇
D
S
and
by
Frobenius.
Moreover,
the
Hodge
filtration
on
D
S
induces
a
Hodge
filtration
on
Geo(D
S
).
Observe
that
with
this
extra
data,
Geo(D
S
)
becomes
isomorphic
to
the
uniformizing
MF
∇
-object
associated
to
Φ
log
(of
Definition
1.3
of
Chapter
III).
Now
we
want
to
pass
to
Galois
representations.
Let
us
assume
that
we
have
chosen
(once
and
for
all)
a
base-point
of
S
that
avoids
the
divisors
defining
the
log
structure.
In
the
following,
our
fundamental
groups
will
be
with
respect
to
this
base-point.
Since
our
construction
will
be
canonical,
we
can
work
étale
locally
on
S.
Thus,
we
can
assume
that
S
is
affine.
We
may
also
assume
that
S
log
is
small
(in
the
sense
of
[Falt],
§2):
that
is,
S
log
is
log
étale
over
A[T
1
,
.
.
.
,
T
d
]
(with
the
log
structure
given
by
the
divisor
T
1
·
.
.
.
·
T
d
).
We
shall
call
these
parameters
T
1
,
.
.
.
,
T
d
small
parameters.
Then
we
would
like
to
consider
the
ring
B
+
(S
log
)
of
[Falt],
§2.
We
will
not
review
the
definition
of
this
ring
here,
since
it
is
rather
involved,
but
roughly
speaking,
it
is
obtained
by
log
(1)
taking
the
normalization
S
of
S
log
in
the
maximal
covering
of
S
K
which
is
étale
in
characteristic
zero;
(2)
reducing
S
modulo
p
and
taking
its
perfection;
(3)
taking
the
Witt
ring
with
coefficients
in
this
perfection;
(4)
adjoining
the
divided
powers
of
a
certain
ideal
to
this
Witt
ring;
and
(5)
finally,
completing
with
respect
to
a
certain
topology.
In
particular,
(1)
B
+
(S
log
)
is
obtained
as
the
inverse
limit
of
a
projective
system
of
PD-
thickenings
of
the
O
S
F
p
-algebra
S
F
p
;
(2)
B
+
(S
log
)
has
an
ideal
I
+
⊆
B
+
(S
log
)
which
is
Galois-invariant
and
such
that
B
+
(S
log
)/I
+
∼
=
S
∧
(i.e.,
the
p-adic
completion
of
S).
157
Moreover,
B
+
(S
log
)
comes
equipped
with
a
natural
Frobenius
action
(which
we
shall
denote
log
by
means
of
a
superscripted
“F
”),
as
well
as
a
continuous
π
1
(S
K
)-action,
which
commutes
+
log
with
the
Frobenius.
The
Frobenius
invariants
of
B
(S
)
are
given
by
Z
p
⊆
B
+
(S
log
).
There
is
a
Galois
equivariant
injection
β
:
Z
p
(1)
→
B
+
(S
log
).
Frequently,
we
shall
abuse
notation
and
write
β
∈
B
+
(S
log
)
for
the
element
of
B
+
(S
log
)
obtained
by
applying
β
to
some
generator
of
Z
p
(1).
Then
the
Frobenius
action
on
β
takes
β
to
p
·
β.
We
will
denote
by
B(S
log
)
the
ring
obtained
from
B
+
(S
log
)
by
inverting
β
and
p.
This
completes
our
review
of
B
+
(−).
Now
let
us
return
to
the
specific
situation
we
have
at
hand.
By
thinking
of
(D
S
,
∇
D
S
)
as
a
crystal,
and
using
the
fact
that
B
+
(S
log
)
is
an
inverse
limit
of
PD-thickenings
of
a
certain
O
S
F
p
-algebra,
one
can
evaluate
this
crystal
on
B
+
(S
log
)
(and
complete
p-adically)
to
obtain
a
B
+
(S
log
)-module
which
we
shall
denote
by
D
S
⊗
O
S
B
+
(S
log
)
(where
the
“hat”
denotes
p-adic
completion).
Alternatively,
one
can
embed
O
S
into
B
+
(S
log
)
by
means
of
a
choice
of
small
parameters,
and
then
take
the
literal
tensor
prod-
uct,
as
described
in
[Falt],
§2.
In
our
situation,
however,
since
we
are
given
a
Frobenius
lifting
Φ
log
,
the
most
useful
point
of
view
will
be
to
embed
O
S
→
B
+
(S
log
)
O
S
into
B
+
(S
log
)
by
means
of
the
Frobenius
lifting
Φ
log
.
Indeed,
the
choice
of
Frobenius
lifting
gives
us
an
embedding
of
O
S
into
the
ring
of
Witt
vectors
that
appears
in
the
construction
(reviewed
above)
of
B
+
(S
log
).
Then,
we
may
regard
the
module
considered
above
as
obtained
via
the
literal
tensor
product
with
respect
to
this
particular
embedding
of
O
S
into
B
+
(S
log
).
At
any
rate,
D
S
⊗
O
S
B
+
(S
log
)
has
a
natural
filtration
and
Frobenius
action.
Let
T
log
→
S
log
be
the
finite
covering
defined
by
Φ
log
.
(Thus,
T
log
∼
=
S
log
.)
Then
log
D
S
⊗
O
S
B
+
(S
log
)
also
has
a
natural
action
by
π
1
(T
K
).
The
reason
why
we
must
restrict
log
),
is
that
the
way
the
Galois
action
is
defined
to
T
log
,
rather
than
considering
all
of
π
1
(S
K
(see
[Falt],
§2)
involves
exponentiating
the
connection
∇
D
S
,
so
in
order
for
the
exponential
series
to
converge,
one
must
be
in
a
situation
where
the
connection
acts
in
a
sufficiently
log
)
nilpotent
fashion.
For
convenience,
let
us
write
Π
T
log
(respectively,
Π
S
log
)
for
π
1
(T
K
log
(respectively,
π
1
(S
K
)).
Let
us
recall
the
uniformizing
Galois
representation
P
et
(Definition
1.4
of
Chapter
III)
associated
to
the
ordinary
Frobenius
lifting
Φ
log
.
Recall
that
P
et
fits
into
an
exact
sequence
of
Π
T
log
-modules
0
→
Θ
et
Φ
(1)
→
P
et
→
Z
p
→
0
The
space
of
splittings
of
this
sequence
then
forms
an
affine
Z
p
-scheme,
which
is
the
∨
∨
).
More
concretely,
Aff(P
et
)
is
a
polynomial
ring
over
Z
p
in
spectrum
of
some
ring
Aff(P
et
158
3g
−
3
+
r
variables
which
is
equipped
with
an
action
by
Π
T
log
.
Moreover,
the
submodule
∨
∨
∨
→
Aff(P
et
).
We
shall
refer
to
Aff(P
et
)
as
the
of
polynomials
of
degree
≤
1
is
given
by
P
et
∨
affinization
of
P
et
.
∨
∨
Note
that
Aff(P
et
)
has
a
natural
Π
T
log
-invariant
augmentation
Aff(P
et
)
→
F
p
.
Let
Gal
∨
D
S
be
the
p-adic
completion
of
the
PD-envelope
of
Aff(P
et
)
at
this
augmentation.
Thus,
D
S
Gal
is
equipped
with
a
natural
structure
of
Π
T
log
-algebra,
and,
moreover,
we
have
a
Π
T
log
-invariant
inclusion
∨
→
D
S
Gal
P
et
In
other
words,
D
S
may
be
identified
with
the
ring
of
additive
periods
(of
Definition
1.5
of
Chapter
III).
On
the
other
hand,
since
P
et
is
the
Galois
representation
contravariantly
associated
to
Geo(D
S
),
it
follows
that
we
have
a
morphism
P
et
→
Geo(D
S
)
∨
⊗
O
S
B
+
(S
log
)
which
respects
the
Hodge
filtrations,
Frobenius
actions
(where
P
η
is
endowed
with
the
trivial
Hodge
filtration
and
Frobenius
action),
and
Galois
actions
(by
Π
T
log
).
“Switching
duals,”
we
thus
see
that
we
have
a
morphism
∨
Geo(D
S
)
→
P
et
⊗
Z
p
B
+
(S
log
)
→
D
S
Gal
⊗
Z
p
B
+
(S
log
)
which
respects
Hodge,
Frobenius,
and
Galois.
Next,
since
Geo(D
S
)
generates
D
S
as
a
“PD-polynomial
algebra”
with
no
relations,
it
thus
follows
that
we
obtain
a
morphism
D
S
→
D
S
Gal
⊗
Z
p
B
+
(S
log
)
which
respects
Hodge,
Frobenius,
and
Galois.
Finally,
tensoring
with
B
+
(S
log
),
we
obtain
the
following
result:
Proposition
1.2.
We
have
a
morphism
D
S
⊗
O
S
B
+
(S
log
)
→
D
S
Gal
⊗
Z
p
B
+
(S
log
)
which
respects
Hodge
filtrations,
Frobenius
actions,
and
Galois
actions
(by
Π
T
log
).
The
Crystalline-Induced
Galois
Representation
Let
U
log
⊆
X
log
be
a
small
affine
open
subset.
Choose
a
Frobenius
lifting
Ψ
log
on
U
log
that
is
compatible
with
Φ
log
on
S
log
.
Thus,
Ψ
log
gives
us
an
embedding
of
O
U
into
B
+
(U
log
)
which
fits
into
a
commutative
diagram
159
O
S
⏐
⏐
−→
B
+
(S
log
)
⏐
⏐
O
U
−→
B
+
(U
log
)
We
would
like
to
consider
∨
⊗
O
U
B
+
(U
log
)
G
U
=
E
D
def
where
the
“∨”
denotes
the
dual
as
a
D
S
-module.
The
problem
is
to
show
that
G
U
has
enough
Frobenius-invariant
sections
in
the
zeroth
step
of
its
filtration.
The
reason
that
we
cannot
apply
the
theory
of
[Falt],
§2
directly
is
that
the
relevant
theorem
(Theorem
2.4
of
loc.
cit.)
assumes
a
bound
on
the
number
of
steps
in
the
filtration
of
the
MF
∇
-object
under
consideration.
On
the
other
hand,
E
D
has
infinitely
many
steps,
What
we
can
do
is
base-change
G
U
by
the
morphism
of
Proposition
1.2.
We
then
obtain
a
free
(D
S
Gal
⊗
Z
p
B
+
(U
log
))-module
G
U
of
rank
two.
Moreover,
G
U
is
equipped
with
a
Galois
action,
a
Hodge
filtration,
and
a
Frobenius
action
F
∗
((G
U
)
F
)
∼
=
G
U
(where
the
superscripted
“F
”
denotes
base-change
by
the
Frobenius
on
B
+
(U
log
)).
Proposition
1.3.
The
submodule
(F
0
(G
U
))
F
=1
(consisting
of
Frobenius-invariant
elements
of
F
0
(G
U
))
forms
a
free
D
S
Gal
-module
E
U
Gal
of
rank
two.
Proof.
The
proof
is
entirely
the
same
as
that
of
Theorem
2.4
of
[Falt],
§2.
The
point
of
base-changing
by
the
morphism
of
Proposition
1.2
is
that
this
enables
us
to
replace
∨
whose
Hodge
filtrations
have
infinitely
many
steps
by
objects
like
G
U
objects
like
E
D
whose
Hodge
filtration
has
essentially
only
two
steps.
In
fact,
over
T
log
,
the
relative
connection
∇
E
on
E
actually
extends
to
a
full
connection
∇
modulo
p,
so
(E
F
p
,
∇
)
defines
a
Galois
representation
onto
some
F
p
-vector
space
E
(⊆
E
F
∨
p
⊗
B
+
(U
log
)
F
p
)
of
dimension
two.
Moreover,
G
U
has
a
filtration
(defined
by
taking
divided
powers
of
the
augmentation
160
ideal
of
D
S
Gal
→
F
p
)
whose
subquotients
are
tensor
products
of
E
F
∨
p
⊗
B
+
(U
log
)
F
p
with
symmetric
powers
of
Ω
et
(−1)
F
p
.
That
is
to
say,
we
know
that
F
0
(−)
F
=1
for
all
of
these
subquotients
is
as
desired,
so
next
we
want
to
consider
the
issue
of
whether
the
various
extensions
involved
split
over
B
+
(U
log
).
But
this
issue
is
precisely
that
discussed
in
the
proof
of
Theorem
2.4
of
[Falt],
§2.
Thus,
we
see
that
we
have
enough
Frobenius
invariants,
at
least
over
the
PD-completion
of
D
S
Gal
.
But
it
is
a
simple
exercise
to
see
that
the
fact
that
the
original
Frobenius
action
is
defined
over
D
S
Gal
(i.e.,
not
just
over
the
PD-completion
of
D
S
Gal
)
implies
that
the
Frobenius
invariants
will
also
be
defined
over
D
S
Gal
itself.
This
completes
the
proof.
def
Let
X
T
log
=
X
log
×
S
log
T
log
.
Similarly,
we
have
U
T
log
⊆
X
T
log
.
Now
we
have
a
natural
π
1
((U
T
log
)
K
)-action
on
G
U
Since
this
action
preserves
the
filtration
and
commutes
with
Frobenius,
we
thus
get
an
action
of
π
1
((U
T
log
)
K
)
on
E
U
Gal
,
which
is
compatible
with
mul-
tiplication
by
elements
of
D
S
Gal
and
the
Π
T
log
-action
on
D
S
Gal
.
Moreover,
as
we
vary
the
open
subset
U
⊆
X,
the
resulting
E
U
Gal
’s
are
clearly
compatible.
Thus,
they
glue
together
to
form
a
π
1
((X
T
log
)
K
)-D
S
Gal
-module
Gal
E
X
We
state
this
as
a
Theorem:
Theorem
1.4.
The
crystalline-induced
MF
∇
-object
(E
D
;
F
i
(E
D
);
Φ
E
D
;
∇
E
D
)
of
Theorem
1.1
has
associated
to
it
an
(up
to
±1)
π
1
((X
T
log
)
K
)-D
S
Gal
-module
Gal
E
X
Gal
which
is
a
free
D
S
Gal
-module
of
rank
two.
We
shall
refer
to
E
X
as
the
crystalline-induced
∇
Galois
representation
associated
to
the
induced
MF
-object
of
Theorem
1.1.
Remark.
Unlike
the
case
of
canonical
curves,
where
one
actually
has
a
dual
crystalline
representation
(in
the
sense
of
[Falt],
§2)
into
GL
±
2
(Z
p
),
in
the
case
of
noncanonical
curves,
Gal
E
X
is
as
close
a
p-adic
analogue
as
one
can
get
to
the
canonical
representation
in
the
complex
case.
In
the
following
subsection,
we
shall
make
the
phrase
“as
close
as
one
can
get”
more
precise.
161
Relation
to
the
Canonical
Affine
Coordinates
log
Let
T
∞
→
S
log
be
the
inverse
limit
of
the
coverings
obtained
by
iterating
Φ
log
.
Let
log
log
log
be
the
pull-back
to
T
∞
of
X
log
→
S
log
.
Let
us
choose
base-points
once
and
X
T
∞
→
T
∞
for
all,
and
let
def
Π
1
=
π
1
((X
T
log
)
K
);
def
Π
∞
=
π
1
((X
T
log
)
)
∞
K
Thus,
Π
∞
⊆
Π
1
is
a
closed
subgroup.
In
the
preceding
subsection,
we
constructed
a
D
S
Gal
-
Gal
Π
1
-module
which
we
called
E
X
.
If
we
restrict
our
Galois
representations
from
Π
1
to
Π
∞
,
then
we
obtain
a
Π
∞
-equivariant
surjection
π
S
:
D
S
Gal
→
Z
p
whose
kernel
is
the
augmentation
ideal
I
S
Gal
,
i.e.,
the
PD-ideal
generated
by
Ω
Gal
S
.
Thus,
if
Gal
we
base
change
E
X
by
π
S
,
we
obtain
a
rank
two
Z
p
-module
with
a
continuous
Π
∞
-action
which
we
denote
by:
def
Gal
⊗
D
Gal
,π
S
Z
p
E
0
=
E
X
S
Thus,
in
summary,
E
0
has
the
advantage
that
it
is
of
rank
two
over
Z
p
,
but
the
disadvantage
Gal
that
it
only
has
a
Π
∞
-
(as
opposed
to
a
full
Π
1
-)
action,
while
E
X
has
the
advantage
that
is
has
a
natural
Π
1
-action,
but
the
disadvantage
that
it
is
of
rank
two
over
the
rather
large
ring
of
additive
periods
D
S
Gal
.
In
this
subsection,
we
show
that
the
canonical
affine
coordinates
(of
Chapter
III,
Theorem
3.6)
measure
precisely
the
degree
to
which
the
Π
∞
action
on
E
0
cannot
be
extended
to
a
full
action
of
Π
1
.
We
begin
with
the
following
fundamental
observation.
Let
Δ
⊆
Π
∞
⊆
Π
1
be
the
log
).
Let
geometric
fundamental
group,
i.e.,
the
kernel
of
the
natural
morphism
Π
1
→
π
1
(T
K
us
consider
the
exact
sequence
of
Π
∞
-modules
Gal
⊗
D
Gal
D
S
Gal
/(I
S
Gal
)
2
→
E
0
→
0
0
→
E
0
⊗
Z
p
I
S
Gal
/(I
S
Gal
)
2
→
E
X
S
By
considering
the
underlying
Δ-module
structures,
we
obtain
that
the
above
exact
se-
quence
defines
an
extension
class
η
Gal
∈
H
1
(Δ,
Ad(E
0
))
⊗
Z
p
I
S
Gal
/(I
S
Gal
)
2
=
H
1
(Δ,
Ad(E
0
))
⊗
Z
p
Ω
Gal
S
log
which
is
fixed
by
the
natural
action
of
π
1
((T
∞
)
K
)
on
this
cohomology
group.
On
the
other
hand,
because
our
original
indigenous
bundle
is
ordinary,
we
see
that
we
have
a
log
)
K
)-equivariant
inclusion
π
1
((T
∞
162
∨
1
(Ω
Gal
S
)
→
H
(Δ,
Ad(E
0
))
Then
we
claim
that
η
Gal
is
precisely
the
class
in
H
1
(Δ,
Ad(E
0
))
⊗
Z
p
Ω
Gal
that
corresponds
S
to
this
inclusion.
Indeed,
this
follows
immediately
from
observing
that
η
Gal
is
essentially
the
Galois
version
of
the
class
η
E
(the
obstruction
to
the
existence
of
a
full
connection
on
E)
that
we
encountered
on
our
way
to
constructing
E
D
.
It
then
follows
immediately
from
the
way
one
passes
from
MF
∇
-objects
to
Galois
representations
that
η
Gal
is
the
above
inclusion,
as
claimed.
This
observation
concerning
η
Gal
will
be
the
fundamental
“hard
fact”
underlying
what
we
do
in
this
subsection;
the
rest
will
be
general
nonsense.
The
general
nonsense
that
we
will
use
is
the
theory
of
[Schl].
Let
us
denote
the
Π
∞
-module
E
0
⊗
Z
p
F
p
by
(E
0
)
F
p
.
Note
that
since
over
T
log
,
the
obstruction
to
putting
a
full
connection
on
E
F
p
vanishes,
so
we
get
a
genuine
MF
∇
-object
(whose
underlying
vector
bundle
is
(E
D
⊗
D
S
/I
S
)
F
p
)
modulo
p.
Thus,
the
Π
∞
-action
on
(E
0
)
F
p
extends
to
a
natural
(dual
crystalline)
action
of
Π
1
on
(E
0
)
F
p
.
We
apply
Schlessinger’s
theory
to
the
functor
on
artinian
rings
B
with
residue
field
F
p
that
assigns
to
such
a
ring
B
the
set
of
isomorphism
classes
of
continuous
representations
of
Δ
on
a
free
B-module
E
B
of
rank
two
such
that
(E
B
)
⊗
B
F
p
=
(E
0
)
F
p
.
Since
H
i
(Δ,
Ad((E
0
)
F
p
))
is
zero
if
i
=
1,
and
of
dimension
6g
−
6
+
2r
over
F
p
if
i
=
1,
it
follows
easily
from
[Schl]
that
this
functor
is
prorepresented
by
a
formal
scheme
R
over
Z
p
.
Moreover,
R
is
formally
smooth
over
Z
p
,
of
relative
dimension
6g
−
6
+
2r.
Now
we
claim
that
there
is
a
natural
continuous
action
of
π
1
((T
log
)
K
)
on
R.
Indeed,
let
α
∈
Π
1
.
Since
Δ
⊆
Π
1
is
a
normal
subgroup,
for
any
representation
of
Δ
on
some
E
B
as
above,
we
obtain
a
new
representation
by
conjugating
elements
of
Δ
by
α,
and
then
acting
on
E
B
in
the
original
fashion.
Since
the
original
Δ-action
on
E
B
extends
to
a
full
Π
1
-action
on
(E
B
)
⊗
B
F
p
,
it
follows
that
this
new
representation
is
isomorphic
to
the
old
after
base
change
by
B
→
F
p
.
Thus,
the
new
representation
defines
a
new
B-valued
point
of
R.
This
defines
an
action
of
Π
1
on
R
which
is
clearly
trivial
on
Δ
⊆
Π
1
.
Thus,
we
obtain
a
natural
action
of
π
1
((T
log
)
K
)
on
R.
Gal
Let
us
now
turn
to
applying
R
to
understanding
the
Π
1
-module
E
X
.
Let
D
S
Gal
be
Gal
the
p-adic
completion
of
D
S
Gal
.
The
underlying
Δ-module
structure
on
E
X
defines
a
classifying
morphism
κ
:
Spf(
D
S
Gal
)
→
R
(This
is
O.K.
despite
the
fact
that
D
S
Gal
⊗
Z
p
Z/p
N
Z
is
not
artinian,
since
Δ
is
topologically
finitely
generated.)
Let
σ
0
:
Spf(Z
p
)
→
R
be
the
composite
of
Spf(π
S
)
with
κ.
Let
R
PD
be
the
p-adic
completion
of
the
PD-envelope
of
R
at
σ
0
.
Let
163
κ
PD
:
Spf(
D
S
Gal
)
→
R
PD
be
the
morphism
induced
by
κ.
Then
the
morphism
induced
by
κ
PD
on
the
Zariski
tangent
spaces
at
σ
0
is
precisely
the
injection
corresponding
to
the
class
η
Gal
considered
above.
Thus,
we
have
the
following:
Lemma
1.5.
The
morphism
κ
PD
is
a
closed
immersion
of
formal
schemes.
Gal
actually
comes
from
Now
let
us
consider
the
fact
that
the
Δ-module
structure
on
E
X
Gal
a
Π
1
-module
structure
which
is
D
S
-semilinear
(with
respect
to
the
Π
1
-action
on
D
S
Gal
log
through
π
1
(T
K
)).
If
we
translate
this
statement
by
means
of
the
functorial
interpretation
of
R,
we
obtain
the
following:
log
log
)-equivariant
with
respect
to
the
natural
π
1
(T
K
)-
Lemma
1.6.
The
morphism
κ
is
π
1
(T
K
Gal
actions
on
D
S
and
R.
log
Now
let
φ
:
Γ
→
π
1
(T
K
)
be
a
continuous
homomorphism
of
topological
groups.
Let
1
→
Δ
→
Π
Γ
→
Γ
→
1
be
the
pull-back
of
the
group
extension
log
1
→
Δ
→
Π
1
→
π
1
(T
K
)
→
1
by
means
of
φ.
Then
one
can
consider
the
issue
of
whether
or
not
the
Δ-action
on
E
0
extends
to
a
continuous,
Z
p
-linear
action
of
Π
Γ
on
E
0
.
Note
that
since
H
0
(Δ,
Ad(E
0
))
=
0,
as
long
as
we
require
that
the
associated
determinant
representation
of
Π
Γ
is
the
cyclotomic
character,
such
an
extension
will
always
be
unique
(up
to
±1).
On
the
other
hand,
by
the
same
reasoning
as
that
used
in
Lemma
1.6,
the
Δ-action
on
E
0
will
extend
to
a
Π
Γ
-action
on
E
0
if
and
only
if
the
Z
p
-valued
point
σ
0
of
R
is
fixed
by
Γ
(acting
through
φ).
Moreover,
by
the
preceding
two
lemmas,
we
see
that
σ
0
is
fixed
by
Γ
if
and
only
if
the
Z
p
-valued
point
of
D
S
Gal
defined
by
π
S
is
stabilized
by
Γ.
But
this,
in
turn,
is
equivalent
to
the
statement
that
the
restriction
of
the
canonical
extension
class
η
Φ
(discussed
just
before
Definition
1.5
of
Chapter
III)
becomes
trivial
when
restricted
to
Γ.
Now
let
us
suppose
that
(B,
m
B
)
is
a
local
ring
with
residue
field
k
which
is
p-adically
complete,
Z
p
-flat,
and
has
a
topologically
nilpotent
PD-structure
on
m
B
.
(Note
that
for
such
a
ring
B,
log
:
(1+m
B
)
→
m
B
and
exp
:
m
B
→
(1+m
B
)
define
inverse
isomorphisms.)
Let
Γ
=
π
1
(Spf(B)
K
).
Suppose
further
that
ψ
:
Spf(B)
→
S
164
is
a
morphism
whose
image
avoids
the
divisor
defining
the
log
structure
on
S
log
.
Then
for
log
)
which
is
some
closed
subgroup
Γ
⊆
Γ
of
finite
index,
we
have
a
morphism
φ
:
Γ
→
π
1
(T
K
compatible
with
the
morphism
induced
on
fundamental
groups
by
ψ.
Recall
from
Chapter
III,
Theorem
3.8,
the
canonical
affine
coordinates
corresponding
to
ψ.
These
coordinates
are
valued
in
B
(or,
more
precisely,
in
m
B
).
Moreover,
they
are
zero
if
and
only
if
the
class
η
Φ
becomes
zero
when
restricted
to
Γ
.
It
is
easy
to
see
that
η
Φ
|
Γ
=
0
if
and
only
if
η
Φ
|
Γ
=
0.
Also,
we
know
from
Chapter
III
that
ψ
corresponds
to
a
canonical
curve
if
and
only
if
the
canonical
affine
coordinates
are
zero.
Thus,
putting
everything
together,
we
obtain
the
following
result:
Theorem
1.7.
The
morphism
ψ
corresponds
to
a
canonical
curve
if
and
only
if
the
Δ-action
on
E
0
extends
to
a
Z
p
-linear,
continuous
action
of
Π
Γ
on
E
0
.
More
generally,
but
less
precisely,
we
see
that:
(*)
The
canonical
affine
coordinates
in
m
B
associated
to
ψ
are
a
mea-
sure
of
the
obstruction
to
extending
the
Δ-action
on
E
0
to
a
Z
p
-linear,
continuous
action
of
Π
Γ
on
E
0
.
Since
the
class
η
Φ
is
“as
nonzero
as
it
can
be”
on
S
log
,
we
thus
see
that
we
have
justified
Gal
the
statement
made
at
the
end
of
the
preceding
subsection
that
the
Π
1
-module
E
X
is
“as
close
as
one
can
get”
to
extending
the
Δ-action
on
E
0
to
a
full
Π
1
-action.
Remark.
In
some
sense,
we
can
describe
what
we
have
done
in
this
subsection
as
follows.
Consider
the
obstruction
to
extending
the
Δ-action
on
E
0
to
an
action
of
Π
Γ
.
A
priori,
this
obstruction
is
highly
nonlinear
and
difficult
to
get
one’s
hands
on
explicitly.
Note
that
this
nonlinearity
exists
despite
the
fact
that
we
already
have
a
Π
∞
-action
on
E
0
,
and
the
discrepancy
between
Π
1
and
Π
∞
is
“essentially”
a
linear
Z
p
-space
of
rank
3g−3+r.
Rather,
the
nonlinearity
arises
fundamentally
from
the
fact
that
we
are
considering
representations
into
GL
±
2
,
which
is
not
an
abelian
(or
even
solvable)
group.
In
particular,
the
moduli
space
R
of
representations
of
Δ
into
GL
±
2
has
no
natural
linear
structure.
Thus,
the
point
of
Gal
Gal
constructing
E
X
and
reinterpreting
the
existence
of
E
X
in
terms
of
R,
as
we
have
done
in
this
subsection,
was
to
linearize
this
obstruction
by
means
of
the
uniformization
of
(the
relevant
part
of)
R
PD
by
means
κ
PD
.
Remark.
For
the
reader
interested
in
pursuing
analogies
with
the
complex
case,
we
also
make
the
following
observation.
Since
R
is
the
local
moduli
space
of
deformations
of
the
canonical
representation
Δ
→
GL
±
2
(Z
p
)
of
the
geometric
fundamental
group,
it
is
natural
to
regard
R
as
a
sort
of
local
p-adic
analogue
of
the
space
R
C
of
isomorphism
classes
of
representations
of
the
geometric
fundamental
group
into
PSL
2
(C)
in
the
complex
case.
Thus,
R
C
has
complex
dimension
6g
−
6
+
2r.
Inside
R
C
,
one
has
Fricke
space
R
R
,
with
real
dimension
6r
−
6
+
2r,
corresponding
to
the
representations
into
PSL
2
(R).
In
a
neighborhood
of
the
canonical
representation
of
a
curve,
R
R
maps
diffeomorphically
onto
165
M
g,r
.
Thus,
one
can
regard
the
subspace
Spf(
D
S
Gal
)
→
R
PD
as
analogous
to
Fricke
space
R
R
⊆
R
C
in
the
complex
case.
The
Parabolic
Case
Before
proceeding,
we
pause
to
take
a
brief
look
at
what
happens
for
elliptic
curves
(regarded
parabolically).
Although
there
are
many
similarities
between
the
parabolic
and
hyperbolic
cases,
there
are
also
certain
differences.
This
is
not
so
surprising
if
one
considers
the
canonical
representations
arising
from
uniformizations
in
the
complex
case.
Indeed,
in
the
complex
context,
for
hyperbolic
curves,
the
canonical
representation
of
the
funda-
mental
group
into
PSL
2
(R)
is
completely
well-defined
up
to
conjugation
by
an
element
of
PSL
2
(R),
while
for
elliptic
curves,
the
morphism
(induced
by
deck
transformations
on
the
uniformization
by
C)
gives
a
representation
of
the
fundamental
group
into
the
group
of
translations
G
a
(C)
of
the
complex
plane,
but
this
representation
is
not
well-defined
up
to
conjugation
by
an
element
of
G
a
(C).
Rather,
there
is
an
ambiguity
of
multiplication
by
a
complex
number.
On
the
other
hand,
this
same
phenomenon
of
“lack
of
rigidity”
ultimately
is
a
con-
sequence
of
the
overall
linearity
of
the
situation,
which
has
positive
aspects,
as
well.
For
instance,
one
can
carry
out
the
construction
of
the
induced
MF
∇
-object
(Theorem
1.1)
for
elliptic
curves
just
as
in
the
hyperbolic
case.
However,
precisely
because
the
obstruction
to
defining
an
MF
∇
-object
is
entirely
linear
from
the
outset,
this
approach
is
a
sort
of
overkill.
Thus,
in
the
following
we
propose
to
examine
the
obstruction
to
defining
a
“full”
MF
∇
-object
(i.e.,
the
same
obstruction
as
the
one
we
examined
in
the
hyperbolic
case)
directly,
at
the
level
of
Galois
representations,
without
resorting
to
the
tool
of
crystalline
induction.
Thus,
let
us
assume
that
f
log
:
X
log
→
S
log
is
an
ordinary
zero
pointed
curve
of
genus
one
such
that
the
classifying
morphism
S
→
M
1,0
is
étale.
Since
the
canonical
representation
should
be
an
extension
of
a
rank
one
étale
representation
by
its
dual
Tate
log
log
twisted
once,
we
consider
the
local
system
R
1
(f
K
)
et,∗
Z
p
(1)
on
S
K
.
Let
us
denote
the
def
log
)-module
corresponding
to
this
local
system
by
H
1
.
Then
there
exists
a
Π
S
=
π
1
(S
K
rank
one
Π
S
-submodule
E(1)
⊆
H
1
such
that
the
action
of
Π
S
on
E
is
unramified,
and
we
have
an
exact
sequence:
0
→
E(1)
→
H
1
→
E
∨
→
0
Suppose
that
E
is
an
étale
Π
S
-module
such
that
(E
)
⊗2
=
E
∨
.
Then,
the
canonical
representation
of
the
universal
elliptic
curve
should
be
an
extension
of
E
by
(E
)
∨
(1).
If
we
tensor
the
above
exact
sequence
by
E,
we
get
an
exact
sequence
of
Π
S
-modules
0
→
E
⊗2
(1)
→
H
1
⊗
E
→
Z
p
→
0
Thus,
the
obstruction
to
the
existence
of
such
a
canonical
representation
is
precisely
the
obstruction
to
lifting
1
∈
Z
p
in
the
above
exact
sequence.
This
obstruction
class
lies
in
166
H
1
(Π
S
,
E
⊗2
(1))
and
coincides
with
the
class
η
Φ
defined
by
the
canonical
Frobenius
lifting
Φ
log
on
S
log
.
That
is
to
say,
we
end
up
with
essentially
the
same
conclusion
as
in
the
hyperbolic
case:
Namely,
that
the
obstruction
to
the
existence
of
a
“canonical
representation”
for
the
universal
log
))
is
given
precisely
by
the
class
η
Φ
defined
ordinary
elliptic
curve
(defined
on
all
of
π
1
(X
K
by
the
canonical
Frobenius
lifting.
§2.
Canonical
Objects
Over
the
Stack
of
Multiplicative
Periods
In
this
Section,
we
note
that
by
working
over
the
stack
of
multiplicative
periods,
we
can
construct
all
the
objects
that
we
are
familiar
with
from
the
case
of
canonical
curves.
The
Stack
of
Multiplicative
Periods
ord
Let
S
log
=
(N
g,r
)
log
.
Let
Φ
log
be
the
canonical
Frobenius
lifting
on
S
log
(as
in
Chapter
III,
Theorem
2.8).
If,
for
some
N
≥
1,
we
take
the
N
th
iterate
of
Φ
log
,
we
get
a
finite,
flat
covering
(Φ
log
)
N
:
S
log
→
S
log
of
S
log
.
Let
P
log
be
the
inductive
limit
of
these
coverings
(as
N
goes
to
infinity).
Let
P
log
be
the
p-adic
completion
of
P
log
.
Definition
2.1.
We
shall
call
P
log
(respectively,
P
log
)
the
universal
(respectively,
com-
plete)
stack
of
multiplicative
periods.
Unlike
the
rings
of
additive
periods
considered
earlier,
which,
roughly
speaking,
are
gen-
erated
by
adjoining
the
logarithms
of
the
multiplicative
parameters
to
Z
p
,
the
structure
sheaf
of
the
stack
of
multiplicative
periods
is
obtained
essentially
by
adjoining
all
p-power
roots
of
the
multiplicative
parameters
to
the
structure
sheaf
of
the
original
base
scheme
(or
stack).
More
generally,
let
T
log
be
a
formal
log
scheme,
whose
underlying
scheme
T
is
p-
adically
complete
and
flat
over
Z
p
.
Let
h
log
:
Y
log
→
T
log
be
an
r-pointed
stable
curve
of
genus
g
that
arises
from
some
classifying
morphism
T
log
→
S
log
.
Then
if
we
pull
back
the
morphism
P
log
→
S
log
via
the
classifying
morphism
T
log
→
S
log
for
h
log
,
we
obtain
an
object
P
T
log
→
T
log
167
over
T
log
.
Let
P
T
log
be
the
p-adic
completion
of
P
T
log
.
We
shall
always
assume
that:
log
(*)
The
log
structure
of
T
Q
is
trivial
over
an
open
dense
set.
p
For
instance,
typically
T
log
will
be
the
normalization
of
Z
p
in
a
finite
field
extension
of
Q
p
with
a
log
structure
that
is
trivial
in
characteristic
zero.
Definition
2.2.
We
shall
call
P
T
log
(respectively,
P
T
log
)
the
(respectively,
completed)
formal
scheme
of
multiplicative
periods
associated
to
the
curve
h
log
:
Y
log
→
T
log
.
Note
that
P
T
log
depends
on
the
choice
of
classifying
morphism
T
log
→
S
log
for
the
curve
log
that
lifts
the
morphism
T
log
→
M
g,r
defined
by
the
curve
itself.
That
is,
P
T
log
depends
on
a
choice
of
quasiconformal
equivalence
class
for
h
log
:
Y
log
→
T
log
.
Remark.
Often
in
what
follows
we
shall
work
in
the
universal
case,
that
is,
over
S
log
,
and
thus
obtain
objects
over
P
log
(or
P
log
).
However,
one
should
always
remember
that
the
objects
constructed
define
(by
restriction)
objects
over
P
T
log
(or
P
T
log
)
for
any
r-pointed
stable
curve
of
genus
g
over
T
log
satisfying
the
hypotheses
just
stated.
The
Canonical
Log
p-divisible
Group
ord
log
Let
S
log
=
(N
g,r
)
:
that
is,
the
locus
of
smooth
ordinary
curves
(with
a
choice
of
quasiconformal
equivalence
class).
Thus,
the
log
structure
on
S
log
is
trivial.
Let
f
log
:
X
log
→
S
log
be
the
universal
curve.
Let
f
log
[N
]
:
X
log
[N
]
→
S
log
be
the
pull-back
of
f
log
by
the
N
th
iterate
of
the
canonical
Frobenius.
Thus,
if
we
pull
(E,
∇
E
)
back
to
X
log
[N
]
and
reduce
modulo
p
N
,
the
obstruction
to
defining
a
full
connection
(relative
to
X
log
[N
]
→
Spec(A))
vanishes,
and
so,
we
obtain
an
MF
∇
-object,
which
we
shall
call
E[N
].
Alternatively,
this
MF
∇
-object
can
be
obtained
taking
the
MF
∇
-object
in
Theorem
1.1,
pulling
back
to
X
log
[N
],
reducing
modulo
p
N
,
and
then
reducing
modulo
I
S
N
.
By
[Falt],
Theorem
7.1,
away
from
the
divisor
at
infinity,
and
the
method
of
Chapter
IV,
§2
(following
[Kato2])
at
the
divisor
at
infinity,
E[N
]
defines
a
finite,
flat
log
group
object
G[N
]
on
X
log
[N
].
Let
P
log
→
S
log
be
the
stack
of
multiplicative
periods.
Let
f
log
[∞]
:
X
log
[∞]
→
P
log
be
the
pull-back
of
f
log
to
P
log
.
Then
by
restricting
from
X
log
[N
]
to
X
log
[∞]
and
then
taking
the
inductive
limit,
we
obtain
an
(up
to
±1)
log
p-divisible
group
G[∞]
=
lim
G[N
]|
X
log
[∞]
−→
on
X
log
[∞].
Definition
2.3.
We
shall
call
G[∞]
the
canonical
log
p-divisible
group
on
X
log
[∞].
168
If
we
invert
p,
and
pass
to
p-adic
Tate
modules,
then
we
obtain
an
étale
local
system
of
Z
p
⊕
Z
p
’s
(up
to
±1)
on
X
log
[∞]
Q
p
.
Now
in
the
notation
of
§1,
we
have
Π
∞
=
π
1
(X
log
[∞]
Q
p
)
Thus,
the
p-adic
Tate
module
of
G[∞]
Q
p
is
given
by
the
representation
of
Π
∞
on
E
0
which
was
discussed
in
§1.
Let
us
denote
this
representation
by
ρ
∞
:
Π
∞
→
GL
±
(E
0
)
Definition
2.4.
We
shall
call
ρ
∞
the
canonical
Galois
representation
of
Π
∞
.
ord
Remark.
If
T
is
any
p-adically
complete
formal
scheme
which
is
Z
p
-flat,
and
φ
:
T
→
N
g,r
is
a
morphism,
then
even
though
the
canonical
log
p-divisible
group
and
the
canonical
Galois
representation
are
not
defined
until
one
goes
up
to
the
scheme
of
multiplicative
ord
periods,
one
can
nonetheless
pull-back
the
canonical
indigenous
bundle
(E,
∇
E
)
N
on
N
g,r
to
obtain
an
indigenous
bundle
(E,
∇
E
)
T
on
X
T
log
→
T
.
This
indigenous
bundle
is
defined
over
T
,
i.e.,
one
needn’t
pass
to
the
scheme
of
multiplicative
periods.
Definition
2.5.
We
shall
call
(E,
∇
E
)
T
the
canonical
indigenous
bundle
on
X
T
log
.
One
should
always
remember
that
one
only
obtains
the
canonical
indigenous
bundle
after
ord
(i.e.,
a
quasiconformal
equivalence
class)
of
the
classifying
choosing
a
lifting
φ
:
T
→
N
g,r
morphism
T
→
M
g,r
of
the
curve.
The
Canonical
Frobenius
Lifting
We
continue
with
the
notation
of
the
preceding
subsection.
It
follows
from
Chapter
IV,
Proposition
3.2,
that
the
supersingular
divisor
D
⊆
X
F
p
(where
f
log
:
X
log
→
S
log
is
the
universal
curve)
is
étale
over
S
F
p
.
We
shall
denote
its
complement
in
X,
the
ordinary
locus
of
X
log
,
by
X
ord
.
Now
it
is
immediate
that
the
construction
of
the
canonical
Frobenius
lifting
over
the
ordinary
locus
(preceding
Theorem
1.6
of
Chapter
IV)
carries
over
immediately
to
the
present
case
(where
the
base
is
S,
as
opposed
to
the
ring
of
Witt
vectors
with
coefficients
in
a
perfect
field).
Thus,
if
we
denote
base-change
by
Φ
log
(the
canonical
Frobenius)
by
means
of
a
superscript
“F
,”
we
obtain
the
universal
analogue
of
Theorem
1.6
of
Chapter
IV:
Theorem
2.6.
There
exists
a
unique
ordinary
Frobenius
lifting
(called
canonical)
log
ord
)
→
((X
log
)
ord
)
F
Φ
log
X
:
(X
169
over
the
ordinary
locus
such
that
we
get
a
horizontal
morphism
Φ
∗
X
E
F
→
E
which
preserves
the
Hodge
filtration.
Now
let
T
be
any
p-adically
complete
formal
scheme
which
is
Z
p
-flat,
and
let
φ
:
T
→
ord
N
g,r
be
a
morphism.
Let
X
T
log
→
T
be
the
pull-back
of
the
universal
curve
by
φ.
Write
X
T
log
F
for
the
pull-back
of
the
universal
curve
by
Φ
◦
φ.
Then
by
restricting
the
morphism
log
Φ
X
of
Theorem
2.7,
we
obtain
a
T
-morphism
log
ord
)
T
→
(X
log
)
ord
Φ
log
T
F
X
T
:
(X
Thus,
in
the
spirit
of
Definition
4.18
of
Chapter
IV,
we
make
the
following:
log
Definition
2.7.
We
shall
call
X
T
log
F
the
Frobenius
conjugate
curve
to
X
T
.
We
shall
call
log
Φ
log
X
T
the
p-adic
Green’s
function
of
the
curve
X
T
→
T
.
Next,
we
consider
compactifications
of
this
canonical
Frobenius
lifting.
Let
D[∞]
⊆
X[∞]
F
p
be
the
result
of
base-changing
to
P
.
Let
D[∞]
⊆
X[∞]
be
the
respective
p-adic
completions.
Let
X
D
[∞]
be
the
completion
of
X[∞]
at
D[∞].
Then
just
as
in
§3
of
Chapter
IV,
by
looking
at
the
universal
deformation
spaces
of
the
canonical
log
p-divisible
group
and
its
double
Frobenius
conjugate,
we
obtain
an
isomorphism
Ψ
:
X
D
F
2
[∞]
∼
=
X
D
[∞]
Let
Y
[∞]
→
X[∞]
be
the
finite,
flat
covering
(of
degree
p
+
1)
parametrizing
cyclic
subgroups
(in
the
Drinfeldian
sense)
of
the
canonical
log
p-divisible
group.
Then,
just
as
before,
there
exists
a
divisor
E
⊆
Y
[∞]
F
p
that
maps
isomorphically
onto
D[∞]
F
p
.
Let
Y
E
[∞]
be
the
completion
of
Y
[∞]
at
E.
Then
we
obtain
an
embedding
(D
E
,
R
E
)
:
Y
E
[∞]
→
X
D
[∞]
×
P
X
D
F
2
[∞]
The
image
of
this
embedding
is
a
divisor,
which,
when
restricted
to
the
ordinary
locus,
is
equal
to
the
union
of
the
graph
of
the
canonical
Frobenius
on
(X
log
)
ord
with
its
“Ψ-
transpose,”
as
in
Chapter
IV,
§3
(see
the
discussion
preceding
Definition
3.3).
Definition
2.8.
We
shall
call
Y
[∞]
→
X[∞],
together
with
Ψ
and
(D
E
,
R
E
)
the
com-
pactification
of
the
canonical
Frobenius
lifting
Φ
log
X
.
Thus,
although
the
canonical
Frobenius
lifting
Φ
log
X
is
defined
over
S
(without
passing
to
the
stack
of
multiplicative
periods),
the
compactification
is
only
defined
over
the
completed
stack
of
multiplicative
periods.
170
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172
Index
Terminology
Note:
The
data
following
each
term
indicate
the
chapter
and
section
where
the
term
first
appears.
admissible
compactified
Frobenius
lifting
admissible
Frobenius
lifting
admissible
indigenous
bundle
associated
Schwarz
structure
IV.4
IV.4
II.2
I.1
Beltrami
differential
Beltrami
equation
Bers
embedding
bianalytic
constant
function
uniformizer
biformal
function
Intro.1
Intro.1
Intro.1
I.1
I.1
I.1
I.1
canonical
affine
parameter
canonical
bilinear
form
canonical
class
canonical
(crystalline)
Galois
representation
canonical
curve
canonical
Frobenius
lifting
(modular)
canonical
Frobenius
lifting
(on
a
curve)
canonical
Galois
extension
class
canonical
height
canonical
indigenous
bundle
canonical
lifting
(point)
canonical
local
Hecke
correspondence,
n
th
canonical
local
systems
canonical
log
p-divisible
group
canonical
multiplicative
parameter
III.1
II.3,
III.3
III.3
IV.1,
V.2
III.3
III.2
IV.1,
V.2
III.1
I.2
Intro.2,
V.2
III.1
IV.3
III.1,
III.3
IV.2,
V.2
III.1,
III.3
173
canonical
p-divisible
group
canonical
pseudo-Hecke
correspondence,
n
th
canonical
representation
(complex
case)
canonical
uniformization
compactification
of
a
Frobenius
lifting
compactification
of
the
canonical
Frobenius
lifting
crystalline-induced
Galois
representation
crystalline
induction
crystalline
Schwarz
structure
with
nilpotent
monodromy
III.1
IV.3
Intro.1,
Intro.2
III.3
IV.4
V.2
V.1
V.0
I.1
I.1
differential
local
system
dilatation
divisor
of
marked
points
III.1,
III.3
Intro.1
I.1
elliptic
Riemann
surface
Intro.1
Fenchel-Nielsen
coordinates
finite
type
(Riemann
surface
of)
F
L-bundle
Frobenius
conjugate
curve
Frobenius
lifting
Frobenius
on
R
1
f
∗
τ
X
log
/S
log
Intro.1
Intro.1
II.1
V.2
III.1
II.2
generic
uniformization
number
global
n
th
canonical
Hecke
correspondence
Green’s
function
(complex
case)
Green’s
function,
p-adic
II.3
IV.3
Intro.1
IV.4,
V.2
Hasse
invariant
Hecke
operators
Hecke
type
height
of
a
Frobenius
lifting
Hilbert
transformation
II.2,
II.3
IV.3
IV.3
IV.4
Intro.1
174
Hodge
section
hyperbolic
metric
hyperbolic
Riemann
surface
hyperbolic
volume
I.2
Intro.1
Intro.1
Intro.1
Igusa
curve
indigenous
bundle
of
restrictable
type
indigenous
section
induced
MF
∇
-object
infinitesimal
Verschiebung
intrinsic
bundle
II.3
Intro.2,
I.2
III.2
V.1
II.2
I.2
Jacobson’s
formula
II.2
Kähler
metrics
and
Frobenius
liftings
Kodaira-Spencer
morphism
Intro.2
I.1,
I.2
local
compactification
of
a
Frobenius
lifting
local
height
of
a
Frobenius
lifting
locally
intrinsic
bundle
locally
stable
of
dimension
one
log
admissible
covering
log
p-divisible
group
IV.4
IV.4
I.2
I.1
III.2
IV.2
marked
MF
∇
-object
I.1
II.2
naive
compactification
of
a
Frobenius
lifting
nilpotent
indigenous
bundle
normalized
P
1
-bundle
with
connection
IV.4
II.2
I.1
175
I.2
ordinary
curve,
hyperbolically
parabolically
ordinary
Frobenius
lifting
ordinary
indigenous
bundle
ordinary
locus
II.3
II.3
III.1
II.3
IV.1
p-curvature
parabolic
Riemann
surface
parabolic
structure
parabolic
volume
element
partition
curves
pre-Schwarz
structure
pseudo-uniformizer
II.1
Intro.1
I.1
Intro.1
Intro.1
I.1
I.1
quasiconformal
equivalence
class,
p-adic
quasiconformal
function
quasidisk
III.3
Intro.1
Intro.1
real
Riemann
surface
renormalized
Frobenius
pull-back
ring
of
additive
periods
Intro.1
III.2
III.1
scheme
of
multiplicative
periods
Schwarz
structure
Schwarzian
derivative
Serre-Tate
parameter
singular
small
affine
small
parameters
smooth
and
unmarked
stable
curve
stack
of
multiplicative
periods
supersingular
divisor
V.2
I.1
I.1
IV.1
I.1
V.1
V.1
I.1
I.2
V.2
II.2
176
tangential
local
system
Teichmüller
metric
Teichmüller
uniformization
topological
marking,
p-adic
totally
degenerate
curve
trianalytic
constant
function
triformal
function
III.1,
III.3
Intro.1
Intro.1
III.3
I.3
I.1
I.1
I.1
uniformizing
Galois
representation
uniformizing
MF
∇
-object
III.1
III.1
Verschiebung
of
an
indigenous
bundle
log
⊗2
)
(−D)
on
f
∗
(ω
X/S
morphism
II.2
II.2
II.2
Weil-Petersson
inner
product
Weil-Petersson
metric
Wolpert’s
coordinates
of
degeneration
Intro.1
Intro.1
Intro.1
Major
Notation
A
Φ
B
B
μ
B
+
(−)
C
III.1
Intro.1,
II.3
III.3
V.1
I.2,
IV.4
(D,
∇
D
)
Δ
D
E
(D
E
,
R
E
)
III.2
V.1
II.2
V.2
177
d
S
D
S
;
I
S
D
S
Gal
(D
y
,
R
y
)
D
0
(a,
b)
(E
D
,
∇
E
D
)
E
E
Gal
E
X
(E
N
,
∇
E
N
)
η
Φ
E
0
F
∗
I.1
V.1
V.1
IV.3
IV.2
V.1
II.2
V.1
III.2
III.1
V.1
II.2,
III.2
G
g,r
;
G
Δ
G
log
GL
±
G[∞]
H
E
H
Y
log
ht(Φ);
ht
x
(Φ)
H
x
ι
(a,b);0
II.3
IV.2
I.1
V.2
II.2
IV.3
IV.4
IV.3
IV.2
J
κ
κ
A
κ
β
L
S
;
L
×
S
I.1
V.1
III.1
III.1,
III.3
I.1
M
g,r
I.2,
ord
M
g,r
μ
=
(Δ,
)
N
g,r
ord
N
g,r
Ω
et
g,r
Ω
et
N
Ω
et
Φ
Ω
log
S
III.2
II.3
III.3
II.2
II.3,
III.2
II.3
III.3
III.1
V.1
178
×
ω
U
log
/S
log
I.1
O
U
#
I.1
P
(P,
F
·
(P),
∇
P
,
P
Φ
)
P
E
P
et
Φ
E
Φ
log
N
Φ
log
X
Φ
ω
E
Φ
τ
E
Π
1
;
Π
∞
P
log
Ψ
Ψ
αβ
Ψ
x
Q
Q
g,r
q
ω,α
q
ω,z
q
θ
ρ
ρ
∞
II.1
III.1
II.2
III.1
IV.1,
V.1
III.2
IV.1,
V.2
II.2
II.2
V.1
III.3
V.2
V.2
III.1
IV.3
Intro.1
II.2
III.3
II.1
IV.1
IV.1
V.2
S;
S
×
S
g,r
I.1
I.3
adm
S
g,r
T
T
1/2
Θ
E
Θ
et
g,r
Θ
et
N
Θ
et
Φ
Θ
F
∗
Θ
V
E
U
bi
U
bi
U
can
II.2
II.1,
IV.1
IV.1
II.2
II.3
III.3
III.1
III.2
II.2
I.1
I.1
III.1,
III.3
179
U
tr
U
tr
I.1
I.1
V
E
V
g,r
V
μ,β
X(a,
b)
ord
X
ord
;
(X
log
)
ord
X
0
(n)
II.2
II.2
III.3
IV.3
IV.1
IV.2
180